176 [June 19, 



XIV. "On the Differential Coefficients and Determinants of 

 Lines, and their application to Analytical Mechanics." 

 By A. COHEN, Esq. Communicated by Professor STOKES, 

 Sec. U.S. Received May 8, 1862. 



(Abstract.) 



1 . The object of this paper is to develope a new method of proving 

 and extending the formulae of analytical mechanics, and at the same 

 time to show how the different steps themselves, in the analytical work 

 by which those formulae are generally arrived at, exactly correspond to 

 mechanical or geometrical facts, just as it is shown in modern geo- 

 metry that the various equations of analytical geometry are capable 

 of important interpretation. 



2. If OA and OB be two straight lines drawn from the origin O, 

 then for well-known reasons AB maybe called "the complete differ- 

 ence" of OA and OB, and may be denoted by (OB)- (OA). Simi- 

 larly, if OA and OB represent two successive states of a variable line 

 P at times t and t + at, then the line whose direction is the limiting 



direction of AB, and whose magnitude is the limit of , will be 



at 



called " the complete differential coefficient" of P, and will be denoted 

 by D,(P). 



3. It is easy to see that the line which represents, in magnitude 

 and direction, a particle's velocity is the complete differential coeffi- 

 cient of the particle's radius vector, and that, similarly, the line repre- 

 senting the particle's acceleration is the complete differential coeffi- 

 cient of the line representing the velocity, and is therefore the second 

 differential coefficient of the radius vector. 



In this case, therefore, Kinematics may be considered as the cal- 

 culus of the first and second differential coefficients of lines. 



4. The following is the fundamental theorem concerning the dif- 

 ferential coefficients of lines : the differential coefficient of a line is 

 compounded of what would be the differential coefficient if the length 

 alone varied, and of what would be the differential coefficient if the 

 direction alone varied. Supposing the line to move in one and the 

 same plane, it is easy to deduce from that theorem the expression for 

 a line's differential coefficient, and by applying to the component 



