142 Mr. Boots on a certain Multiple Definite Integral. 
in which, for abbreviation, 
r= (a—F-F..) ofan yaa) jwtiz, (5) 
2 
i i : on Ps 
and since the above expression vanishes whenever eras 73 .. transcends the 
limits assigned in (3), it is evident, that on substitution in (2), we may extend 
the limits of the integrations relative to x, 7,... from — © to », and suppose 
them to be taken in the limiting sense above explained. Hence we have 
—- oe fhe \; dadvdw wf(a)). . dx,dxr,..cos(T). (6) 
We shall first consider the expression 
\ 5 UR 5 G08 (@'). 
Now, arranging tT with reference to the suffixes, 
a 2 
x 2 Ge ae 
..dx,..cos(T)=\ ..dz,..cos (av— 4 v—(a,—.2,)?w—...+7=). 
: &: ) \. ( es ) = 5) 
u 
pees v = wi i awh? ih a, vw 
— oS ; ree 
z h? ' vtwh?/] ' vot hfw 
a,wh, 
Let «,— ——+. = u, the limits of u,, are — » and , and substituting in a 
v + wh? 
similar manner for the other variables, we have 
is S a? : v+h2wo = 
ea, -- cos(T) = {au +. COS (a (et )™ +45 _ +a ut... (7) 
Now by successive applications of the theorem, 
2 
s e ™ Tw 
\ du, cos (atru,’) =F Cos (ax ys 
we find 
.. du, ..du,cos(a—ru2.. —7,U,2) = ——™ cos (4 — 7) 
he \ a ) (FT nTn)? (a ™4 (8) 
