476 Mr. E. P. Culverwell. Maxima and Minima [June 10, 



another table, and follows the " march " first indicated by Dr. Hugh- 

 lings Jackson as existing in epileptic seizures. 



This march is in accordance with Axiom I, since the shoulder com- 

 mences the series of movements in the uppermost part of the area, 

 the thumb at the lowest part, and the wrist in the intermediate 

 part. 



Summary. — 1. That X is the superior frontal sulcus of man. 



2. That the movements of the joints are progressively represented 

 in the cortex from above down. 



3. Localisation of sequence of movements. 



4. Localisation of quality of movements. 



5. That there is no absolute line of demarcation between the 

 different centres. 



III. " On the Discrimination of Maxima and Minima Solutions 

 in the Calculus of Variations." By E. P. Culverwell. 

 Communicated by Professor B. Williamson. Received 

 June 5, 1886. 



(Abstract.) 



In the first part of the paper it is shown that the usual investigation 

 by which the second variation of an integral is reduced, requires that 

 the variation given to y (the undetermined function) is such that its 

 differential coefficients, taken with regard to x (the independent 



variable) are continuous up to the twice-wth order, — £ being the 



highest differential coefficient of y appearing in the function to be 

 integrated. Bat it is not necessary that the variation should be con- 

 tinuous beyond its (n — l)th differential coefficient, and a method of 

 reducing the variation to Jacobi's form by a process which is not 

 open to the above objection is then given ; and the method has the 

 additional advantage that its simplicity enables it to be easily 

 extended to other cases where there are more than two variables. 



Bat in dealing with multiple integrals especially, any method 

 depending on algebraic transformation is necessarily defective, inas- 

 much as it is invalid unless solutions, which do not become either 

 zero or infinite within the limits of the integration, can be found for 

 a number of simultaneous partial differential equations containing 

 at least as many unknown quantities as equations. It is pointed out 

 that it is not in general possible to obtain such solutions, and that 

 even when the particular problem is assigned, it would be impracti- 

 cable to ascertain whether there were such solutions. 



The method given in the second part of the paper does not depend 

 on or require any algebraic transformation. The second variation is 



