■ + 



dV 

 dp 



dp 



dx 



+ 



dV 

 dq 



d x 



+ 



dV 



dr 



dr 

 dx 



&c. 



j 

















324 Mr. Cankrlen on the Problems of 



changing the sign of k, and, by properly assuming k, may be 

 made greater than the sum of all the terms which follow it, if 

 U,— U /; be either a maximum or a minimum, it is necessary 

 that N — P + Q" — R'" + &c. = 0. This equation when in- 

 tegrated will give us the relation between x and y, or the 



equation y — F (x) which renders / dxV a maximum or a 



// 

 minimum. 



The order of the differential equation we have just found 

 will in general be 2 n, if the order of the highest differential 

 coefficient in V be n. We may find an equation one order 

 lower in this way : 

 dJV) _ y, _ _dV_ dV dy 



dx d x d y d x 



+ 



or, V — M + Nj9 + P q + Q r + R s + &c. 



but = — p { N — P' + Q" — R'" + Sec.} by the equation 



last found. 



.\V'= M+ Pq+ V'p + Qr-pQ" + Rs + R'"p + &c. 



Now, Pq + p P' = {Pp)' ; Q r - p Q" = Q r + Q'q - 

 Q' q - Q" p = (Q q - p Q')' ; R s + R"' p = R s + r R'- 

 r R' - q R" + q R" + p R m = (R r - q R' + p R")' ; 

 &c. = &c.: substituting these results we find 

 V'=M + {Pp)' + {Qq-pQ'Y + {Rr-qR' + pR")' 

 + &c, and 

 .-.V = (3 + fdxM + Pp + Qq-pQ' + Rr-qR' + 

 pR" + &c. (l). 



This equation contains one arbitrary constant |S : and if the 

 order of the highest differential coefficient in V be n, there will 

 in general be 2 n arbitrary constants in the primitive equa- 

 tion between x and y. These are to be determined by means 

 of the limits and other data which, according to the case, 

 must be granted for that purpose. But if the limits are not 

 given, the constants which are expressed in terms of the 

 symbols representing them, must remain arbitrary, as was 

 observed above. Our object then, in order to complete the 

 solution in this case, must be to determine the actual values 

 of the limits which these symbols represent. 



Since y t = $(x) and y t/ = \J/ (,r /y ) we may consider U / — U /; 

 as a function of x t and x n \ now if we substitute x { -\- lx / for x / 

 and x u + Sx^for x n in U / — U //5 the result, which we will de- 

 note by oo / — cu // , is {J, — U//+ ju.8^4- irZxjj + &c, /x and -rr 

 representing the coefficients of 8 x / and 8 x u in the expanded 

 expression, and since U / — U^ is always greater or always 



less 



