THE COURSE OF MATHEMATICAL STUDIES. 425 



points of view, as a seat of learning and as a place of education. 

 Much may be done by means of prizes to encourage learned men 

 in the pursuit of the kind of knowledge to which they have 

 especially dedicated themselves ; but as things are, and perhaps 

 as they ought to be, even a liberal education must end about 

 the age at which the Bachelor's degree is commonly taken. 

 And it may further be affirmed, with much show of reason, that 

 those whom after that epoch circumstances still permit "inter 

 silvas Academi quaerere verum," may with advantage, so far as 

 the symmetrical development of the mind is concerned, turn 

 from mathematical to other studies. 



V. With respect to the subjects mentioned in the seventh 

 query, I have already made some remarks on the lunar and 

 planetary theories, as well as on electricity and the kindred 

 branches of physics. There remain, therefore, only the calculus 

 of variations and definite and elliptic integrals. Of these sub- 

 jects the first seems not unsuited for University reading. It 

 involves important principles and admits of important applica- 

 tions. From its connexion with the theory of conditions of 

 integrability it forms a natural sequel to the integral calculus ; 

 and, on the other hand, if the planetary theory were to be 

 studied in the manner which I have suggested, some previous 

 acquaintance with its principles would be indispensable. In 

 favour of definite integrals there is not so much to be said, many 

 of them depending for their evaluation on particular artifices, 

 which must uselessly burden the Student's memory ; and if in 

 an examination he attempts to determine the value of one with 

 which he is not already acquainted, he will often only waste 

 time and ingenuity to no purpose. Still certain definite inte- 

 grals must be known, and the theory of definite integrals of 

 periodic functions is especially important from its connexion 

 with Fourier's theorem and the development of discontinuous 

 functions. The difficulties of this theory have, I think, been 

 sufficiently removed to justify its introduction into our course ; 

 and it is not to be forgotten that no other step in the recent 

 progress of analysis has exercised so great an influence on ma- 

 thematical physics. The general theory of elliptic integrals is 

 too extensive and too abstruse for University reading, and the 

 study of isolated propositions almost useless. But Abel's theo- 

 rem ought to be studied as the natural development of the theory 



