216 GENERAL THEOREMS 



where the sign of summation extends to all the quantities F, and 

 where 



(aft-a'ff) cos (a'a!+ffy)+(aff+5a') sin (a' 



From the known integrals 



f +00 cos aaj. ^a TT ^ f +00 asin c.#. dy 



- _ = _ e * aa; - ^ - 2 = 

 ./. a +a /_ a + a 2 



the upper signs to be taken when x is positive, it follows that 

 M = wV"***!' (1 + 1 1 1). 



If x and y are both positive, the bracket becomes 1 + 1 + 1 + 1 

 or 4 ; if a; only be negative, it becomes 1 1 1 + 1 or 0; if. y 

 only be negative, it becomes 1 l + l-l orO; and similarly if 

 both x and y are negative. Thus generally 



or = 



There are, indeed, exceptional cases ; as if y be zero, x being 

 positive, when M = 2-7T 2 e~ aa; , and similarly if x be zero, y being 

 positive ; and again, if x and y are both zero, when M = TT* : 

 but of these, as we are about to multiply M by the element 

 dx dy, it is unnecessary to take account. Therefore, in inte- 

 grating for x and y, we include only positive values of the 

 variables ; and as u and u are not to be greater than h and h t 

 respectively, x and y must be such as not to transgress the 

 inequalities 



mx + ny^ h, mx + n t y S h r 



Thus we find that 



(mx + ny] < y (mx + n t y] e-*-^, 



the limits being given by the two above-written inequalities. 

 It appears, therefore, that the integral (1), when there are two 

 limiting inequalities, is reducible to the sum of a series of double 

 integrals. 



This result is analogous to that which is obtained in the case 

 of the function <f> (mx + ...pz) e~ ax -- cz , in the paper already re- 

 ferred to. 



It remains to determine the form of the quantity F^. This 

 is done at once by multiplying equation (2) by 



(a + am + o^raj (b + an + a^), 



