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THE CANADIAN JOURNAL 
NEW SERIES. 
No. XLVI.—JULY, 1863. 
REMARKS ON THE NEGATIVE INDEX OF A FUNCTION. 
BY THE REV. E. K. KENDALL, 
LATE PROFESSOR OF MATHEMATICS IN TRINITY COLLEGE; TORONTO. 
Ib the consideration of indices, whether used to denote powers of 
numerical or algebraical expressions, or the successive performance 
of some operation or function on a quantity, it is usual in examining 
the meaning of negative or fractional indices to state that ct 2s con- 
venient to assign certain interpretations, because of a certain gene- 
rality which then obtains in the results. In the words of a recent 
author,* ‘“ Experience will prove that the notation here given is 
often convenient, and we may shew that it is not altogether an arbi- 
trary notation but one that naturally presents itself.” It appears 
to me that this, at any rate in the case of negative indices, is an 
inadequate mode of expressing the ground on which these indices 
are interpreted, and that the meaning to be assigned to the index 
is not only one that naturally presents itself, not only not altogether 
arbitrary, but the meaning which must be assigned, exclusive of any 
other meaning, and no more arbitrary than the use of the notation 
for positive integral indices. With respect to fractional mdices even, 
Tam of opinion that the above would be an insufficient account of 
the reasons by which we are led to accept an interpretation of the 
index since it would leave an impression that we are guided rather 
* See Todhunter’s ‘‘ Plane Trigonometry.” 
Vot. VIII. Vv 
