284 JIr. GOODWIN, ON THE PURE SCIENCE 



.-. /(0) = cos (w sin -^ /(O) + sin (^-n- sin -)/(^), 



and Pf(e) = PcosU sin -] /(O) + P sin L sin ^ W f - V 



In this case, therefore, it would seem that the oblique quantity P would be equivalent to two 

 components, one in the original direction, the other inclined at an angle of 60°, the magnitude 



of the former being Pcos (ir sin-j, that of the latter Psin (tt sin-j. But there will be a dis- 

 tinction between this and that of the ordinary formula cos 9 + (- l)-5 sin 9 which is to be observed ; 

 for in this latter case the impossible plane determined by f{9) = (— 1)^ coincides with that 

 determined by f(9) = (- \)K but in the present hypothetical case, we have two impossible direc- 

 tions, one corresponding to 9 = - or f (9) = (- l)i, the other to = Stt or f(9) = - (- !)*• 



3 3 



And therefore that which is analogous to the impossible plane in the ordinary case is an im 

 possible cone whose semi-vertical angle is an angle of 60". 



There will be two impossible cones in like manner belonging to the formula 



/(0) = (-lp=cos-+ (-l)Jsin- 



TT TT 



which, together with the formula (J), are particular instances of the general form, 

 /(e) = (-l)('^° = cos ^^ + (-i)Ssin4^ (C). 



7r 



13. The preceding cases are examples of the general formula f(9) = (- 1)®, where O is some 



function of 9 which = when 9 = 0, and = 1 when 9 = w, the direction or directions for which 



2 /b + 1 

 f(fi) = (- O- are given by = . It may be observed, that all examples must have this 



property in common ; that if we suppose a quantity P to be composed of two others whose direc- 

 tions are the line for which 9=0, and that for which f{9) = (- l)^, and if we call these 

 components c and y respectively, then x and y satisfy the condition 



,T- + y- = P- ; 



and therefore if x and y be regarded as oblique co-ordinates of a point, the locus of that point 



9 



is an ellipse ; in the case of f{9) = (- 1)" this locus becomes a circle. 



l*. It is evidently possible to vary indefinitely the law according to which f{9) shall vary 

 from +1 to — 1, while 9 increases from to tt, even though we confine ourselves to the form 

 /(6) = (— 1)®; and all these laws will express modes in which the affection of a quantity may 

 be diametrically reversed ; I am disposed to look upon most of them as fictitious generalizations 

 which can have no type in the nature of things, just as we might construct a system of geometry 

 of four dimensions which could have nothing real corresponding to it. It may be possible, 

 however, to find some which have not this fictitious character, and which express physical laws. 

 We shall obtain a distinct conception of the manner in which the law expressed by the formula 



fi0) = (~ 0" differs from all others, by observing that if (— l)® expresses the law of change from 

 + to -, the gradual change of affection, as compared with a change in the value of 9, will be 



