4 
PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 
Thus the following laws are indicated :— 
u n = v n X w n .(xi.), 
where v n — xv /l _ 1 — (n — l)v*_ 2 .(xii.), 
w n = yw n _ x — (n — 1 .(xiii.). 
We shall now show that these laws hold good by induction. Assume 
u n+1 = v n+l w n+l = (xv n — nv n _ x ) (yw n — nw n _{). 
Thus u n+1 = xyu n + w?u n _ x — n{yw n v n _ l + xv n w n _ 2 ). 
But by (ix.), substituting for from (xi.) and (xiii.), 
u n+ 1 = xy{v n w n + w(w — l)r w _ 2 ^v ; _ 2 j + n(2n — 1 — x z — y^v^iv,,^ 
— n(n — \)v n _ l iv u _ l — xyn(n — l)v n _ 2 w„_ 2 
+ n(n — 1) {yv n _ x io H _^ + xv n _ 2 w n _ x ). 
— xyv n w n + n^v n _ x w H _ x — n(x 3 + y^v^w,,^ 
-\-n(n— 1) ( yv^w „_ 3 + £CV w _ 2 W„_ 1 ) 
= xyv n w n + nH n _yw n _ x — n{yv n _ l {yu n _ l — n — 1 w„_ 2 ) 
+ xtv n _ 1 (xv )l _ 1 — n — lv„_ 2 j 
= xyv n w n + n 2 v n _ 1 w n _ l — n(yv n _ 1 w !l + xw n _, v„) 
= v u+1 w n+l , as we have seen above. 
Thus, if the theorem holds for u n , it holds for u H+v Accordingly 
U = + * (1 + v -fr + + . . . + +...). 
(xiv.), 
where the vs and w’s are given by (x.), (xii.), and (xiii.). 
1 r” r=° 
It is thus clear that U dx cly consists of a series of which the general 
"7T J /i Jj 
term is 
1 
\n 
where 
1 C _ 
V„ = ^7^ e 
w * = ^4?L e " i '’ w * o!y - 
It remains to find these integrals. 
The general form of v u is given by 
°±( n . ~ 1 > a;W -s i nQi - l)(7t - 2)(n - 3) 4 _ 
9 j -j^ A/ “ 90 9 ia/ LvC. 
(XV.). 
