Solutions of Problems in the Calculus of Variations. 199 



If this be a minimum, we shall have 



dp 



The equation obtained by equating to zero the factor in brackets, 

 which alone gives a value pf p depending on the function /(/r), 

 is identical with that deduced by the Calculus of Variations. 

 This instance may suffice to show that the processes of the Cal- 

 culus of Variations embrace solutions in which the principle of 

 discontinuity is involved. Although, in the solution of the 

 problem mentioned at the beginning of these remarks, the dis- 

 continuity enters in a different manner from that exhibited above, 

 the foregoing reasoning may, I think, be accepted as evidence 

 that no antecedent objection lies against a solution on the 

 ground that it involves discontinuity. I proceed now to con- 

 sider further the question of discontinuity with reference to the 

 other problem. 



For this instance, the Calculus of Variations gives, as is 

 known, the differential equation 



2a,,-(b-f)>/T~P = 0, 



a and b being arbitrary constants. This equation, if it could be 

 exactly integrated, would give a relation between x and y invol- 

 ving three constants, all of which would be necessary for com- 



— «r 



pletely answering the proposed question. Although such an 

 integral is not obtainable, the following exact equation is dedu- 

 cible by integration, as I have shown in the communication before 

 quoted, from the above differential equation, viz. 



s being the arc of the curve, and k an arbitrary constant intro- 

 duced by the integration. Xow it is here to be remarked that 

 this equation proves at once that the curve does not pass through 

 the points A and B, because in that case we should have y = 0, 

 and the right-hand side of the equation would be numerically 

 greater than unity ; which is impossible. The ordmates y , y x 

 at those points must, therefore, have certain unknown values 

 which it is required to find. To accomplish this, it must, first, 

 be admitted that the total enclosing surface of the solid is com- 

 posed of the surface generated by the revolution of the curve, 

 and the surfaces generated bv the revolution of the extreme ordi- 

 nates through A and B. Against this admission no argument 

 can be adduced except an a priori objection, for which no reason 

 has hitherto been assigned, against the principle of discontinuity. 



