DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 57 



,3it *- *- " *- 



o - /<? - ? 5- = ~ 



,i* ,u 



fcq nq 5- = 5-3 j -rf p- ' ^ . 



* 82 3*& -^ 3z 2 3z * 3s 



Multiplying the second of these equations by p, the fourth by q, and then adding all 

 four, the function $ is eliminated, and we have 



a + b + c = 0. 



33. Again, the general theory in a preceding Section (see particularly 4) 

 immediately suggests that -u occurs as an argument of an arbitrary function. This 

 being so, let the variables be considered as transformed to x, y, u, where u now is 

 known, and effect the transformation directly upon the equation 



a + I + c 0. 

 We have, for any function P, 



3P , 3P 7 3P j 7T3 

 ir- ax + ~ ay + -5- dz = dc 

 ox oy on 



dP j f /l> rfP 



= -:- aiC + T~ V + V^ M 



dx dy du 



dP dT dP 1 



so that 



3 d j> <1 

 8b; r6; A (f ' 



_3 rf g d 



dy dy A rfw ' 



1 r? 

 A rfi7' 



We thus have 



a? = d& ~ 2 A" 



^^ df 2^ ^, 

 A^ rf tt s - du \ A 2 A ^ h 



- 4- - to" 4- W") 



3 1 ^ h W ' 



A%d~ A 2 *t 2 dw I A A 3 



whence, adding and remembering that a +b-\- c = 0, it follows that 



d rf 2 2 



\-QT~ 1 = 0. 



A * 



VOL. CXCI. A. 



