358 
Proceedings of the Royal Society of Edinburgh. [Sess. 
not so. In his paper, Ueber der Perioden der elliptischen Integrate , Dorpat, 
1875, H. Bruns gave several formulae which can be used; but the un- 
fortunate choice of an absolute invariant leads to unnecessary complication. 
It will be seen later that by the choice of another absolute invariant 
the formulae take simple and symmetrical forms. 
The formulae given by Schwarz and later mathematicians have been 
developed from the purely mathematical point of view, and suffer from a 
surfeit of ir and imaginaries. Had the mathematician skilled in the theory 
of functions been responsible for our trigonometrical tables and formulae, 
it is possible that, instead of, to give one example, the values of sin (9 being 
tabulated, it might have been ( — ) , so that the simplest trigono- 
“•(?) 
metrical operations would have involved a great deal of unnecessary work 
and writing. These remarks apply in a lesser degree to Legendre’s great 
tables of his elliptic integrals. After computing — J 2 f0d0, he carefully 
0 
multiplied the results by so that in nearly every case of practical 
application the computer has to divide by For such reasons, the 
Lt 
symbols ordinarily used have in this paper been replaced by others, as 
follows : — 
-“1 
by 
w i 
2 
—Vi 
7T 
n i 
2 1 
too 
7T ^ 3 
2 
—^3 
7 r 
n z 
i iw l 
1 i 1 
— kge- 
7 r q 
J5 
With these changes, the fundamental formulae become : — 
n^iv 3 + = — 
7 r 
Jw 1 = 
\/ \/ 6^ 6*2 
1 1-3^ + 5 V/- . . . 
1 1 3 l-3 2 ? + 5g5- . . . 
^3=1. 
(l + 2 2 | + 2 2 \ 6 + . . . ) 
