286 Mr. GOODWIN, ON THE PURE SCIENCE OF MAGNITUDE AND DIRECTION. 



We have here an effect which results from two causes, which separately vary uniformly. 

 Considering equation (D), we may observe that the difference between it and the equation 



pf{e)= p cos e + {- i)i p &\n e (E) 



is this, that, although both give 0=0 and = — as the directions of the components, yet the 



valuesof P/(0) and Pfl—\ which are the same in the latter case (omitting the sign of affec- 

 tion) are different in the former : for in equation {D) 



Pf(0) = a. 



P/g)=M-0^. 



In fact, considering the equations (D) and (E) geometrically, the former represents an ellipse, 

 the latter a circle : the angle will manifestly be the eccentric anomaly. 



My reason for introducing the formula (£>) is that I may remark that, whereas the formula 

 (^E) represents force considered in the light of pure direction, the formula (D) corresponds to 

 a case of polarity. Pure force must, of course, be free from any polarity, that is to say, its 

 absolute magnitude must be the same in whatever direction it acts, the direction will modify the 

 effect, not diminish or increase it ; but there are complicated instances of force in which this is 

 not the case, but in which there is polarity ; for example, in the case of an elastic medium under 

 constraint from the action of the particles of a crystallized body which contains it. Now the 

 formula (D) appears to be exactly calculated to express this kind of force ; to fix our conceptions 

 let an elastic medium have the same properties in all sections parallel to the plane of ,vy, and 

 have polarity in that plane ; consider any one section, and let the properties of this section 

 of the medium be symmetrical about the axes of a; and y, then the origin will be a position 

 of rest for a particle, and if it be disturbed, the force of restitution may be represented by such 

 a formula as (Z)). 



In examining that formula we find that there are two directions perpendicular to each other, 

 for which the force of restitution is in the direction of displace- 

 ment ; for all other displacements the force of restitution is not 

 in the same direction, but will have to be determined thus ; let 

 JP be the direction of displacement : take APB proportional to 

 a + b, and AD, making the same angle with Ax as AB, propor- 

 tional to a - b; complete the parallelogram ABCD, and draw 

 through P a line parallel to the diagonal AC; this will be the 

 direction of the force of restitution. Hence then it appears, that 

 the formula (D) will represent the kind of law which determines 

 the force of restitution on a disturbed particle in the case of uniaxal crystals. 



H. GOODWIN. 



