154 Dr. WHEWELL'S SECOND MEMOIR ON THE INTRINSIC EQUATION 



48. If we have s = any series of integral powers of 0, we shall still have a curve of this 

 cycloidal kind, as to its finite part. 



Thus let s = any integral function of [n + 1) dimensions of (p. 



ds 

 Then — = (0 - a) (0 - /3) (0 - 7) &c. to n terms. 



And there is a cusp when = a, when = /3, when = y : 



And in the sinuses, s is alternately positive and negative. 



With regard to the infinite branches, 



ds 

 If s = a<p n+l , we have — — = (n + l) ad> n 

 dm 



And it is evident that if n be positive, - — goes on increasing as increases, that is, as s 



increases. Hence there is an infinite diverging spiral 



Therefore if s = any integral fraction of <p, of (n + l) dimensions, the curve is a cycloidal 

 formed curve having n cusps, and at each end an infinite diverging spiral. Fig. 9. 



ds 



49. If n be negative, — — goes on diminishing as increases, that is, as s increases, if 



n + l be positive. In this case we have a converging infinite spiral. 



ds 

 If n + 1 be negative, (as well as n), — goes on increasing as (f> decreases, that is, as s 



increases; wherefore we have a diverging infinite spiral. • 



When n is negative and n + 1 positive, n is a negative proper fraction, and n + 1 a positive 

 proper fraction : and if we get rid of fractional indices by involution, we shall have (p = some 

 integral power of s ; which /equation, it appears, gives a converging spiral. 



Let (p = any series of integral powers of s. Then we shall have 



(p = (s - a) (s - b) (« - c) &c. 



<p will be alternately positive and negative with maxima and minima, and the curve will be 

 a sinuous curve in the neighbourhood of the values s = a, 6, c &c, and will have a converging 

 spiral at each end. Fig. 10. 



These resemble the running pattern curves <p = m sin s + n sin 2s, &c, except that they 

 have not a cycle of recurrence, and have spiral tails. 



50. The running pattern curve, noticed Art. 17, &c of the former memoir, (of which the 

 equation is cp = m sin s) may be further examined. 



As we have remarked, Art. 34, to find the rectangular co-ordinates of this curve, we have 



/■cos (p d(p r sin <p d<p 



* " J x/wj^T?' V ~ * \Zrri' - (p* ' 

 - cos d>dd> , (b , 



/7^r^ = ^ +v/m, -^ 2( ^ +c ^ +jD ^ 5+&c - ); 



