388 
eho y-s 
Again, if m = 4, the singular solution of 
dw dw dw dw \t dw \t dw \? 
4 de yp ea }(«s2) + (0) + (e%) 
is 
Finally, it is easily Pamet that the singular solution of the partial dif 
ferential equation 
ae. dw 7 solos, , i dw (dw _ 9 
da \" de *! dy dy * * deh" dy y ies g 
is the paraboloid 
Generalesation of Charpit’s Method. 
10. Let us suppose that the given non-linear partial differential equa- 
fion is reduced to the form 
s=f (2, Y, 2, 2,4 1; #). 
Then we have the two conditions 
dr 7) = (4. dt \ _ = | 
dy dx Ws Nady] 
Now on supposition that r is expressed generally as a function of z, y, z, 
p, and g, we have 
— + 
dy) dy da!” dp’ a 
and 
= _ ds ds Le ds ds oe a dr 
dz ~ Ge dz? Pare 7 dx E aie dq” 
ds { dt s dt dé at 
daz ary dp dq 
