The Tensor Formula
(Monday, April 11; 11.30 a. m.)
Markham sat staring at Vance like a man hypnotized. Heath stood rigid,
his mouth partly open, his cigar held a few inches from his lips.
There was something almost comic in the Sergeant’s attitude, and I had
a nervous inclination to laugh; but for the moment my blood seemed
frozen, and all muscular movement was impossible.
Markham was the first to speak. Jerking his head backward, he brought
his hand down violently on the desk-top.
“What new lunacy of yours is this?” He was fighting desperately
against Vance’s dumbfounding suggestion. “I’m beginning to think the
Robin case has affected your mind. Can’t a man with the commonplace
name of Sprigg be shot without your trying to turn it into some
grotesque hocus-pocus?”
“Still, you must admit, Markham old dear,” returned Vance mildly,
“this particular Johnny Sprigg was shot with ‘a little gun’, through
‘the middle of his wig’, so to speak.”
“What if he was?” A dull flush had crept into Markham’s face. “Is that
any reason for your going about babbling Mother-Goose rhymes?”
“Oh, I say! I never babble, don’t y’ know.” Vance had dropped into a
chair facing the District Attorney’s desk. “I may not be a thrillin’
elocutionist; but really, now, I don’t babble.” He gave Heath an
ingratiating smile. “Do I, Sergeant?”
But Heath had no opinion to express. He still held his astonished
pose, though his eyes had now become mere slits in his broad,
pugnacious face.
“Are you seriously suggesting——?” began Markham; but Vance interrupted
him.
“Yes! I’m seriously suggesting that the person who killed Cock Robin
with an arrow has vented his grim humor upon the hapless Sprigg.
Coincidence is out of the question. Such repetitive parallels would
knock the entire foundation out from all sanity and reason. ’Pon my
soul, the world is mad enough; but such madness would dissipate all
science and rational thinking. Sprigg’s death is rather hideous; but
it must be faced. And however much you may force yourself to protest
against its incredible implications you’ll eventually have to accept
them.”
Markham had risen, and was pacing nervously up and down.
“I’ll grant there are inexplicable elements in this new crime.” His
combativeness had gone, and his tone had moderated. “But if we assume,
even tentatively, that some maniac is at large reconstructing the
rhymes of his nursery days, I can’t see how it will help us. It would
practically close all routine lines of investigation.”
“I shouldn’t say that, don’t y’ know.” Vance was smoking meditatively.
“I’m inclined to think that such an assumption would supply us with a
definite basis of inquiry.”
“Sure!” snapped Heath with ponderous sarcasm. “All we gotta do is to
go out and find one bug among six million people. A cinch!”
“Don’t let the fumes of discouragement overcome you, Sergeant. Our
elusive jester is a rather distinctive entomological specimen.
Moreover, we have certain clews as to his exact habitat. . . .”
Markham swung round. “What do you mean by that?”
“Merely that this second crime is related to the first not only
psychologically, but geographically. Both murders were committed
within a few blocks of each other,—our destructive demon at least has
a weakness for the neighborhood in which the Dillard house is
situated. Furthermore, the very factors of the two murders preclude
the possibility of his having come from afar to give rein to his
distorted humor in unfamiliar surroundings. As I learnedly pointed out
to you, Robin was translated into the Beyond by some one who knew all
the conditions obtaining at the Dillard house at the exact hour the
grisly drama was performed; and surely it’s obvious that this second
crime could not have been so tidily staged had not the impresario been
acquainted with Sprigg’s ambulat’ry intentions this morning. Indeed,
the entire mechanism of these weird playlets proves that the operator
was intimately cognizant of all the circumstances surrounding his
victims.”
The heavy silence that followed was broken by Heath.
“If you’re right, Mr. Vance, then that lets Sperling out.” The
Sergeant made even this qualified admission reluctantly; but it showed
that Vance’s argument had not been without its effect on him. He
turned desperately to the District Attorney. “What do you think we’d
better do, sir?”
Markham was still battling against the acceptance of Vance’s theory,
and he did not answer. Presently, however, he reseated himself at his
desk and drummed with his fingers upon the blotter. Then, without
looking up, he asked:
“Who’s in charge of the Sprigg case, Sergeant?”
“Captain Pitts. The local men at the 68th-Street Station grabbed it
first; but when the news was relayed to the Bureau, Pitts and a couple
of our boys went up to look into it. Pitts got back just before I came
over here. Says it’s a washout. But Inspector Moran* told him to stay
with it.”
* Inspector William M. Moran, who died two years ago, was, at the
time of the Bishop case, the Commanding Officer of the Detective
Bureau.
Markham pressed the buzzer beneath the edge of his desk, and Swacker,
his youthful secretary, appeared at the swinging door that led to the
clerical room between the District Attorney’s private office and the
main waiting-room.
“Get Inspector Moran on the wire,” he ordered.
When the connection had been made he drew the telephone toward him and
spoke for several minutes. When he had replaced the receiver, he gave
Heath a weary smile.
“You’re now officially handling the Sprigg case, Sergeant. Captain
Pitts will be here presently, and then we’ll know where we stand.” He
began looking through a pile of papers before him. “I’ve got to be
convinced,” he added half-heartedly, “that Sprigg and Robin are tied
up in the same sack.”
Pitts, a short, stocky man, with a lean, hard face and a black
tooth-brush moustache, arrived ten minutes later. He was, I learned
afterwards, one of the most competent men in the Detective Division.
His specialty was “white-collar” gangsters. He shook hands with
Markham and gave Heath a companionable leer. When introduced to Vance
and me he focussed suspicious eyes on us and bowed grudgingly. But as
he was about to turn away his expression suddenly changed.
“Mr. Philo Vance, is it?” he asked.
“Alas! So it seems, Captain,” Vance sighed.
Pitts grinned and, stepping forward, held out his hand.
“Glad to meet you, sir. Heard the Sergeant speak of you often.”
“Mr. Vance is helping us unofficially with the Robin case, Captain,”
explained Markham; “and since this man Sprigg was killed in the same
neighborhood we thought we’d like to hear your preliminary report on
the affair.” He took out a box of Corona _Perfectos_, and pushed it
across the desk.
“You needn’t put the request that way, sir.” The Captain smiled, and
selecting a cigar held it to his nose with a kind of voluptuous
satisfaction. “The Inspector told me you had some ideas about this new
case, and wanted to take it on. To tell you the truth, I’m glad to get
rid of it.” He sat down leisurely, and lighted his cigar. “What would
you like to know, sir?”
“Let us have the whole story,” said Markham.
Pitts settled himself comfortably.
“Well, I happened to be on hand when the case came through—a little
after eight this morning—and I took a couple of the boys and beat it
up-town. The local men were on the job, and an assistant Medical
Examiner arrived the same time I did. . . .”
“Did you hear his report, Captain?” asked Vance.
“Sure. Sprigg was shot through the top of the head with a .32. No
signs of a struggle—no bruises or anything. Nothing fancy. Just a
straight shooting.”
“Was he lying on his back when found?”
“That’s right. Stretched out nice and pretty, right in the middle of
the walk.”
“And wasn’t his skull fractured where he’d fallen on the asphalt?” The
question was put negligently.
Pitts took his cigar from his mouth and gave Vance a sly look.
“I guess maybe you fellows over here do know something about this
case.” He nodded his head sagaciously. “Yes, the back of the guy’s
skull was all bashed in. He sure had a tough fall. But I guess he
didn’t feel it—not with that bullet in his brain. . . .”
“Speaking of the shot, Captain, didn’t anything about it strike you as
peculiar?”
“Well . . . yes,” Pitts admitted, rolling his cigar meditatively
between his thumb and forefinger. “The top of a guy’s head isn’t where
I’d ordinarily look for a bullet-hole. And his hat wasn’t touched,—it
must have fallen off before he was potted. You might call those facts
peculiar, Mr. Vance.”
“Yes, Captain, they’re dashed peculiar. . . . And I take it the pistol
was held at close range.”
“Not more’n a couple of inches away. The hair was singed round the
hole.” He made a broad gesture of inconsequence. “Still and all, the
guy might have seen the other fellow draw the gun, and ducked forward,
spilling his hat. That would account for his getting the shot at close
range in the top of the head.”
“Quite, quite. Except that, in that case, he wouldn’t have fallen over
back, but would have pitched forward on his face. . . . But go on with
the story, Captain.”
Pitts gave Vance a look of crafty agreement, and continued.
“The first thing I did was to go through the fellow’s pockets. He had
a good gold watch on him and about fifteen dollars in bills and
silver. So it didn’t look like a robbery—unless the guy that shot him
got panicky and beat it. But that didn’t seem likely, for there’s
never any one round that part of the park early in the morning; and
the walk there dips under a stone bluff, so that the view is cut off.
The bird that did the job certainly picked a swell place for it. . . .
Anyhow, I left a couple of men to guard the body till the wagon came
for it, and went up to Sprigg’s house in 93rd Street,—I’d got his name
and address from a couple of letters in his pocket. I found out he was
a student at Columbia, living with his parents, and that it was his
habit to take a walk in the park after breakfast. He left home this
morning about half past seven. . . .”
“Ah! It was his habit to promenade in the park each morning,” murmured
Vance. “Most interestin’.”
“Even so, that don’t get us anywheres,” returned Pitts. “Plenty of
fellows take an early constitutional. And there was nothing unusual
about Sprigg this morning. He wasn’t worried about anything, his folks
told me; and was cheerful enough when he said good-bye to ’em.—After
that I hopped up to the university and made inquiries; talked to a
couple of the students that knew him, and also to one of the
instructors. Sprigg was a quiet sort of chap. Didn’t make friends and
kept pretty much to himself. Serious bird—always working at his
studies. Stood high in his classes, and was never seen going around
with Janes. Didn’t like women, in fact. Wasn’t what you’d call
sociable. From all reports he was the last man to get in a mess of any
kind. That’s why I can’t see anything special in his getting shot. It
must have been an accident of some kind. Might have been taken for
somebody else.”
“And he was found at what time?”
“About quarter of eight. A bricklayer on the new 79th-Street dock was
cutting across the embankment toward the railway tracks, and saw him.
He notified one of the post officers on the Drive, who phoned in to
the local station.”
“And Sprigg left his home in 93rd Street at half past seven.” Vance
gazed at the ceiling meditatively. “Therefore he would have had just
enough time to reach this point in the park before being killed. It
looks as if some one who knew his habits was waiting for him. Neatness
and dispatch, what? . . . It doesn’t appear exactly fortuitous, does
it, Markham?”
Ignoring the jibe Markham addressed Pitts.
“Was there nothing found that could possibly be used as a lead?”
“No, sir. My men combed the spot pretty thoroughly, but nothing showed
up.”
“And in Sprigg’s pockets—among his papers. . . ?”
[Illustration: A torn scrap of paper showing two typewritten
equations: “Bikst = λ/3 (gik gst - gis gkt)”, and
“Bikst = 0 (flat at ∞)”.]
“Not a thing. I’ve got all the stuff at the Bureau—a couple of
ordinary letters, a few odds and ends of the usual kind. . . .” He
paused as if suddenly remembering something, and pulled out a
dog-eared note-book. “There was this,” he said unenthusiastically,
handing a torn, triangular scrap of paper to Markham. “It was found
under the fellow’s body. It don’t mean anything, but I stuck it in my
pocket—force of habit.”
The paper was not more than four inches long, and appeared to have
been torn from the corner of an ordinary sheet of unruled stationery.
It contained part of a typewritten mathematical formula, with the
lambda, the equals and the infinity sign marked in with pencil. I
reproduce the paper here, for, despite its seeming irrelevancy, it was
to play a sinister and amazing part in the investigation of Sprigg’s
death.
Vance glanced only casually at the exhibit, but Markham held it in his
hand frowning at it for several moments. He was about to make some
comment when he caught Vance’s eye; and, instead, he tossed the paper
to the desk carelessly with a slight shrug.
“Is this everything you found?”
“That’s all, sir.”
Markham rose.
“We’re very grateful to you, Captain. I don’t know what we’ll be able
to make out of this Sprigg case, but we’ll look into it.” He pointed
to the box of _Perfectos_. “Put a couple in your pocket before you
go.”
“Much obliged, sir.” Pitts selected the cigars, and placing them
tenderly in his waistcoat pocket, shook hands with all of us.
When he had gone Vance got up with alacrity, and bent over the scrap
of paper on Markham’s desk.
“My word!” He took out his monocle and studied the symbols for
several moments. “Most allurin’. Now where have I seen that formula
recently? . . . Ah! The Riemann-Christoffel tensor—of course! Drukker
uses it in his book for determining the Gaussian curvature of
spherical and homaloidal space. . . . But what was Sprigg doing with
it? The formula is considerably beyond the college curricula. . . .”
He held the paper up to the light. “It’s the same stock as that on
which the Bishop notes are written. And you probably observed that the
typing is also similar.”
Heath had stepped forward, and now scrutinized the paper.
“It’s the same, all right.” The fact seemed to nonplus him. “That’s a
link anyway between the two crimes.”
Vance’s eyes took on a puzzled look.
“A link—yes. But the presence of the formula under Sprigg’s body
appears as irrational as the murder itself. . . .”
Markham moved uneasily.
“You say it is a formula that Drukker uses in his book?”
“Yes. But the fact doesn’t necessarily involve him. The tensor is
known to all advanced mathematicians. It is one of the technical
expressions used in non-Euclidean geometry; and though it was
discovered by Riemann in connection with a concrete problem in
physics,* it has now become of widespread importance in the
mathematics of relativity. It’s highly scientific in the abstract
sense, and can have no direct bearing on Sprigg’s murder.” He sat down
again. “Arnesson will be delighted with the find. He may be able to
work out some astonishing conclusion from it.”
* This expression was actually developed by Christoffel for a
problem on the conductivity of heat, and published by him in 1869 in
the “Crelle Journal für reine und angewandte Mathematik.”
“I see no reason,” protested Markham, “to inform Arnesson of this new
case. My idea would be to keep it under cover as much as possible.”
“The Bishop won’t let you, I fear,” returned Vance.
Markham set his jaw.
“Good God!” he burst out. “What damnable sort of thing are we facing?
I expect every minute to wake up and discover I’ve been living a
nightmare.”
“No such luck, sir,” growled Heath. He took a resolute breath like a
man preparing for combat. “What’s on the cards? Where do we go from
here? I need action.”
Markham appealed to Vance.
“You seem to have some idea about this affair. What’s your suggestion?
I frankly admit I’m floundering about in a black chaos.”
Vance inhaled deeply on his cigarette. Then he leaned forward as if to
give emphasis to his words.
“Markham old man, there’s only one conclusion to be drawn. These two
murders were engineered by the same brain: both sprang from the same
grotesque impulse; and since the first of them was committed by some
one intimately familiar with conditions inside the Dillard house, it
follows that we must now look for a person who, in addition to that
knowledge, had definite information that a man named John Sprigg was
in the habit of taking a walk each morning in a certain part of
Riverside Park. Having found such a person, we must check up on the
points of time, place, opportunity, and possible motive. There’s some
interrelation between Sprigg and the Dillards. What it is I don’t
know. But our first move should be to find out. What better
starting-point than the Dillard house itself?”
“We’ll get some lunch first,” said Markham wearily. “Then we’ll run
out there.”