aSO Mr. GOODWIN, ON THE PURE SCIENCE 



where 3 is any constant angle, the value of which is indifferent so far as the uniformity of 

 variation of f(6') is concerned; and it is clear that the form of / as thus determined satisfies 

 the condition of uniform variation corresponding to the continuous variation of Q. 



Hence, therefore, the formula {A) may be regarded as an expression for the law, according 

 to which change of direction affects any cause, which depends on magnitude and direction only, 

 and which varies uniformly to its direct opposite while its direction varies continuously to the 

 exactly opposite direction. 



5. I shall now interpret the symbolical equation {A) ; we have 



f{d) = cos0 + (- i)i sin^; 



hence, /(O) = 1, 



■^^ ' -J. 



and therefore the equation 

 mav be written thus. 



P/(0) = P cos + ( - l)i P sin e 



P./(0) = Pcose./(O) + Psin0./g) (5). 



From this form it may be concluded, that the equation expresses this fact, that the effect of an 

 oblique cause is measured by that of two whose directions are at right angles to each other, 

 the direction of one being the original direction that of the other the perpendicular to it, and the 

 intensity of the former being measured by P cos Q the intensity of the latter by P sin ; and 

 this comes to the same thing as saying that any cause is equivalent to two other causes, the 

 directions of whose action are perpendicular to each other, and whose intensities are measured 

 by P cos and P sin 6, 6 being the angle which tlie direction of P cos 9 makes with the original 

 direction. 



6. The proposition which I have just been endeavouring to establish is the fundamental 

 one of the pure science of Magnitude and Direction, and may be called tlie principle of the 

 resolution of causes ; from it may be deduced with ease rules for the calculation of the resultant 

 effect of any number of causes, since we have only to resolve them in the same directions and 

 then add the effects ; and if the demonstration given be free from solid objection, we shall have 

 established a proposition which contains within itself the theorems of position, the theorems of 

 the composition of forces, of moments, and of velocity both linear and angular. 



7- As it may appear improbable that so general a proposition as that which I am con- 

 sidering should be capable of proof without reference to particular cases, I am anxious to examine 

 the objections which may be made to the proof just given. But before doing so, I would observe, 

 that it does not seem unreasonable to suppose a priori that it would be possible to establish 

 a formula indicating the variation of intensity of a number of causes, which have been all brought 

 imder the same definition by saying that they are such as depend solely on intensity and direction ; 

 we may consider it as certain that a law which applies to one will apply to all, and the only question 

 is, whether we can establish such a law by reasoning which shall apply equally to all causes 

 comprehended in one definition. Now it is allowed on all hands that the symbols + and - are 

 proper symbols for exactly opposite affections, and therefore it cannot be objected to, that we 

 should consider + P and — P as symbols of two causes of equal intensity but exactly opposite in 

 direction ; the question simply is, according to what law can + P vary continuously and uniformly 



