ee - 
he 
f 
4 ON THE RECENT PROGRESS OF ANALYSIS. 79 
introduced such an independent variable by the assumption = = WX, which 
_ of course implied that ae = — WY. This assumption is unsymmetrical, 
and it is therefore difficult to see how to generalise it. But if we assume 
dz xX d Y 
aa ws we shall of course have 4 m4 Les and therefore ¢ is symme- 
trically related to « andy. Let Fu =0 be an equation whose roots are 
2 and y, then, as we know, whenu = a2, Fu =2—y and, when v= y, 
¥F'u=y — 2, so that using an abbreviated notation 
ae ox vx wnat = vx 
d= Brea dt PG) 
Nothing is easier than to generalise this result. For instance, the “Ja- 
cobische system ” of two equations is 
dz dy dz 
gde ydy z2dz 
hg Bia alle Yar AR 
Now if F xz =0 have a, y, z for its roots, the two preceding equations 
may, in virtue of a very well-known theorem, be replaced by the three follow- 
ing, 
de_ V& dy_ VY dz_VZ 
dt™ Wa’? di” Fly’ dt” Fz’ 
which introduce an independent variable ¢, symmetrically related to a, y and 
2; and so in all cases, 
M. Richelot * then takes a symmetrical function of 2, y,.+. 2, viz. their 
_ sum, and by means of the last written equations arrives at an integrable dif- 
_ ferential equation of the second order, the principal variable being the said 
\ 
__ * As M. Richelot’s method of demonstrating Euler’s theorem is more symmetrical and 
_ far more easily remembered than Lagrange’s, it ought, I think, to be introduced into all 
elementary works on elliptic functions, The equation to be integrated being 
bi aie vit Ah civ Cy ME Wel i op 
Mat Batya pia pia! VatpytyP ty ty | 
assume dv_VYatpaty@piepea 
a dt y—@ 
dy_ VetByt oP PPE, 
: dt @-y 
ne 
_ Letp=x+y. Then after a few obvious reductions 
; dp 1 
- at sp. 
a i. deh. : 
® Gane tr, 
{Vappetye pie pia — Vat pyt PP yryS 
oe =B(y— 2) (y°— 27) + #(y? — 2°), 
the algebraical integral sought. It may easily be expressed in other forms, 
