XXXI 



importance. In another place, we find a sketch in eight pages of a 

 memoir of Legendre's which Dr. Todhunter considers to be the 

 foundation of all that Laplace added in the theories of attraction and 

 the figure of the earth to the works of Maclaurin and Clairaut. As 

 we read the sketch, we see the first beginning of Laplace's coefficients 

 and a recognition of the importance of the potential. This was the 

 commencement of a new era in mathematical physics. In a third 

 place, the history shows us how D'Alembert, trying to find the 

 attraction of an ellipsoid, makes it depend on a single definite 

 integral. This result, Dr. Todhunter reminds ns, is the point at 

 which modern investigations have finally arrived. But D'Alembert, 

 after effecting this, strangely rejects his result as inadmissible. " In 

 his process," says Dr. Todhunter, " there is nothing wrong in prin- 

 ciple, but he has omitted a bracket which renders his result slightly 

 inaccurate. He gives some invalid argument against his method. 

 Thus D'Alembert deliberately rejects one of the most important 

 formulae of the subject, which in fact quite supersedes a large part of 

 his memoir. This is perhaps the strangest of all his strange mis- 

 takes." A little further on in the history, we read how Laplace 

 values and appropriates the treasure which D'Alembert deliberately 

 threw away. 



In 1869, the subject prescribed for the Adams' Prize was "A 

 determination of the circumstances under which Discontinuity of 

 any kind presents itself in the solution of a problem of Maximum 

 or Minimum in the Calculus of Variations." The proposal of this 

 subject seems to have arisen from a controversy which had been 

 carried on in the " Philosophical Magazine " a few years previously. 

 In this controversy Dr. Todhunter had taken part, and when the 

 subject was proposed for the essay he was anxious that his own 

 view should prevail. This view is given in the opening sentences of 

 his essay : " We shall find that, speaking generally, discontinuity is 

 introduced by virtue of some restriction which we impose, either 

 explicitly or implicitly, in the statement of the problems which we pro- 

 pose to solve." This thesis he supports by considering in turn the usual 

 applications of the calculus, and pointing out where he considers the 

 discontinuities which occur to have been introduced into the con- 

 ditions of the problem. This he successfully proves in many 

 instances. In some cases, the want of a distinct test of what dis- 

 continuity is, somewhat obscures the argument. His essay was 

 rewarded with the prize. It is published under the title " Researches 

 in the Calculus of Variations." 



In the midst of so busy a life, Dr. Todhunter could yet find time 

 to write for others. The second edition of Boole's "Differential 

 Equations " was published tinder his care ; and, what is more, he 

 undertook the labour of arranging and editing the supplementary 



