of Edinburgh, Session 1881-82. 
845 
Thus we get 
7ii l SX/xy = — SyyySyX/x , 
namely 
(12) m 1 — — Sy 7l . 
Further 
S<£'X. p,y + S cf>'fx. yX — Sapx/ySaX 4 - S(^ 1 /xy)S/3X 
But 
likewise 
4- SajyXSap. + S/^yXS /3/x 
= S. (Va x y .YXp.. a) + S. (V /? x y .YXp,. (3) 
= S . V Xp [V (aY a l7 ) 4 - Y(fivp iy )] 
— — (SXpa^ay + SX/x^ 1 S/3y)SX/x.y [Saa 1 + S/3(3 l ] . 
Say = SaV a/3 — 0 ; 
S£y-0. 
Thus we get 
(12 bis) m 2 = Saa 1 4 - S /3/3 1 . 
It is clear that these values of m 1 and m 2 are those which would be 
derived from the values of the same coefficients in the case of the 
expression of </>p by three terms instead of by two only. 
Let us now calculate the second member of (9), which for abbre- 
viation sake we will designate by £YXp, putting 
( 13 ) £ = <£ 2 - m 2 <f> 4 - m l . 
We have by (4) and (12 bis) 
(cf> — m 2 ) = paSajp — pSaa x 
+ /3S/? x p - pS/3/3 1 
= Y(Yap.a l ) + Y(V/3p./3 1 ). 
£p = [<£(<£ - m 2 ) + wijp 
ip= aSa 1 [Y(X r ap.a 1 ) + Y(Y^p./5 ] )] 
+ ^S/? 1 [Y(Yap.a 1 ) + Y(V(3p./3 1 )] + m l p. 
Two terms vanish, there remains : 
Then as 
we have 
£p = aS(a x Y f3p.(3^) 4 - /^(/^Yap. a : ) + m Y p 
= - aS./T^pYa^) + /3S.aV(pYa l (3 1 ) + m lP 
— Y.Y (pVa 1 /5 1 )Ya/3 + m Y p 
= Y(Vpy l .y)+m l p 
= p[Syy x + m Y ] - 7l Spy . 
