28 The Rev. R. Murphy o?i a fiew Theorem in Analysis, 



"^fes a, ^, y to the axes of elasticity, of the force put in play 

 on w by a displacement p || it 



= p {Acos^a + Bcos^/S +Ccos^y}: 



2. The sum of such resolved forces in the directions of any 

 three rectangular axes is constant 



= p{A+B + C}: 



3. The force produced by a displacement in the direction 

 of one of the axes of elasticity is the greatest, and in that of 

 another the least that can be produced in any one direction 

 by the same displacement in that direction. 



I am, Gentlemen, yours, &c. 



St. John's College, Cambridge, C. J. 



Aug. 24, 1836. 



VII, On a new Theorem in Analysis, By the 

 Rev. Robert Murphy, M.A. 



T APLACF^ first gave a very simple and elegant demon- 

 ■^^ stration of the theorem generally known as Lagrange's, 

 by taking the partial differential coefficients of u relative to x 

 and a, from the equation y =■ a + x (^ (y), where ti =y {y) ; 

 and the simplicity of the process depends on the manner in 

 which X enters this equation, namely, as a multiplier of a 

 function of i/, 



I have considered a more general equation where x enters 

 the function in any manner, viz. y = a + <^'(j7, y) and u =zf(y), 

 and have obtained the following theorem : 



When X is changed into x + h, in this equation y and 

 consequently u are also changed ; let the latter become U, 

 and let 4> (jt + ^, y) — <^ (^jj/) = A 4> for abridgement; the re- 

 lation between U and u will then be 



da da \^ 1.2 da\ 



^ da^\\,2,S ' daj^^''' 



In proving this we shall use 4> as a contraction of <p {x,y), 

 and any accented symbol will be used to express the partial 



differential coefficient relative to a: ; thus instead of ^^^'^' 



ax 



we shall simply write 4>'. 



Now differentiating the proposed equation y =r a + ^ rela- 



Jative to x and a, we get 



