OF A FUNCTION OF TWO INDEPENDENT VAEIABLES. 235 



Let us now return to equations (1) and (2). Change in 

 < (x, y] the variables x and y to x + h and y + k respectively. 

 Calling u the new value of u, we get 



7 du 7 du h? (d*u 2k d?u k* d*u 

 u=u + hj- + k- r + r- \~r^ + -r jj- + ^ -j- 

 ax dy [2 [ax fi ax ay h a 



Let us now assign to x and y any values consistent with 

 (3), leaving however the ratio of k to h quite arbitrary, and 

 examine whether u' becomes less or greater than u when k 



and h are sufficiently diminished. The coefficient of in 



2. 



the above value of u', is 



<f u 2k d*u ltfu A.M- & 



h + * 



ax n/ ax ay h ay ft n 



Now by (4) this 



A (* , kB \* 



*( l + hAj' 



and is therefore necessarily positive if A be positive, and 

 necessarily negative if A be negative, whatever be the ratio of 

 k to h, except for that particular value of the ratio which makes 

 the expression vanish. Hence the conclusion will be this : if 

 we assign to x and y values consistent with M = 0, then when 

 h and k are sufficiently diminished, u is certainly less than u 



if -T-j be negative, and certainly greater than u if -j- a be 



positive, excepting only when k has to h one particular ratio. 

 This latter case would require further examination, had we 

 not already shewn that by a certain supposition u is reduced to 

 a constant, so that when k has to h the one particular ratio, 

 u is ultimately neither greater than u nor less than M, but 

 equal to it. 



The whole theory may be illustrated geometrically; for 

 example, if 



J3* = a 2 x* y*+ (xcos a+ysina) 2 (1), 



find maxima or minima values of z ; 



