228 MAXIMA AND MINIMA VALUES 



hence $ (a + h, b + k} <f> (a, b) is always of the same sign as 

 A, when h and k are taken small enough, except when 7 is 



K 

 75 



equal to;', and then the sign is as yet unknown and 

 ^L 



further investigation is required. Let P, Q, S, T stand for 

 the values of 



<f u d s u cPu d?u 



dx*' a^d ~~ 



respectively, when x = a and y = b', and let 



x being made = a and y = b after the differentiations. 



Suppose j is equal to -r , then Ah*+ 2Shk + Ck* vanishes, 



1C _il 



and 



<j> (a+h, &+&)-</> (a, 1} = 4 



Hence if h and k be taken small enough the sign of 



<f>(a + h, b + k) -<f>(a, 6) 

 will be the same as the sign of 



Ptf + SQtfk + SShtf + Tk 3 , 



and will therefore change by changing the sign of h and k ; 

 it is impossible then that < (a, 6) can be a maximum or mini- 

 mum value unless 



vanishes when T is equal to r . 

 k A 



Suppose this condition to be satisfied, then the sign of 



-(]>(a, b), 



7 73 



when T is equal to . , is the same as the sign of -R 2 ; and 

 when j- is not equal to -j > an d n an( ^ & are taken small 



