ConwaY—The Partial Differential Equations of Mathematical Physics. 199 
Then if S,>,# is a solution where y is a function of 7 and p, then 
oy i n+ 2p = I Oy < on if n +29 —1 of 
2 
Or” Lr or Op” p Op 
) 
so that 
ws = Ky B(n-2)) ey (B) p). 
In like manner, a general solution of the second equation is 
S, EG, 0-2) (® t). 
VIII.—Kuineric Souurtons. 
All the above results admit of being extended in an important manner. For 
brevity, our remarks will be confined to solutions of Class II., in which the total 
number of variables is even. Writing 
r? — (a a ion) + (av = Oy + se eg 
the integrals are of the form 
j duf(u)(P — (t= uP 
where m is an integer. This function is infinite for the values 
and for an infinite range of values of ¢. ‘To extend this result, let 7), v7... . be 
functions of vw. Then the integral will be taken about a real zero of the function 
r—(t—u); 
and an infinity will occur when, for a zero w of this function, 
D 6 a or a 
el =(¢ =e) |=, O1 Pw UU eV) 
Hence, at an infinity we must have 
—— pul 
~ \ apy)! - 
TRANS. ROY. DUB. SOC., N.S., VOL. VIII., PART XVI. 2N 
