in the Calculus of Variations. 109 



shall now show that the effect of this step is to restrict the gene- 

 rality of the solution. 



Although the above differential equation is not generally inte- 

 grable, we can obtain from it an exact expression for the length 

 s of the curve. For we have 



f?£ _ ds _ 2ay 

 dx dy b — y v 

 and 



. ds= 2a y d v 



\/4ay-{b-y 2 f 

 The integration of this equation gives 





m 



k being a new arbitrary constant. Now at the point A, y = 

 and s=0. Hence 



, 2a 2 + b 2a* + b ,^ 



cos&= ■- — -.- -r—-r==. . , (3) 



2a\/d 2 + b V(2a 2 + 6) 2 -6 2 v '. 



Thus the denominator of this value of cos k is less than the nu- 

 merator ; which is impossible. If s t be the length of the arc, it 



would similarly be found, for the point B, that cos ( — + k ) is 



greater than unity. The inference to be drawn from these results 

 is, that when s = and s = s v y cannot be equal to zero, but must 

 have some other values, which it is required to find before the 

 solution of the question can proceed. This may be done as 

 follows. 



Since the coordinates of the extremities of the arc must have 

 certain values y and y l different from zero, it is necessary that 

 the circular areas generated by the revolution of these coordi- 

 nates should be taken into account in the expression for the total 

 surface of the solid; that is, the reasoning must be conducted 

 in the manner proposed by Mr. Todhunter (p. 410). Hence, if 

 r and r' be distances of points of the circular areas from A and 

 B respectively, there will be the two additional terms, 



8§ardr + S§a? J dr ! , 



the integrations being taken from r = to r = y , and from ^—0 

 to /' = ?/i. Consequently the total quantity outside the sign of 

 integration will be 



