1 38 Sir James Cockle on Primary Forms, 



fy + z—O, a complete integral of (10) will be given by 



the arbitrary constant C^ being added after the f integration, 



and C after the r\ integration. Applying this to the primary 



C ri* 



(8) , we find g= —r= and 17= . Hence C = n gives 



y = t;; and if we take the particular integral \jr(x)=0, we find 



/ a ndoc \ 

 y = P ( 1 J 0. For simplicity add unity after each in- 



tegration. Then, whether we evaluate by series, or by treating 

 y as an integral of the linear equation — , y=0, we 



have y = C +1 (x + %/x 2 + l) n ', and in like manner the second 

 corresponding particular integral may be found. 



7. Boole seems to have considered the regular and the primary 

 forms as of distinct species. This I attribute in part to his 

 not recognizing, in the theory of the former, any change of the 

 independent variable other than that from x to kx n . Under ap- 

 propriate changes there is a certain reciprocity which appears to 

 indicate that all the forms are but varieties of one species. 



8. All binomial {op. cit. p. 430) biordinals may be included in 



d*y 2 a + ex n dy 1 f+gx n 

 dx 2 x h + kx n dx x 2 h + kx nV 

 Taking the criticoid, we have 



d*y L + Mx n + ~Nx 2n _ 

 !x~* + x*(h + kx n )* y ~ ' 



for in this paper I make no explicit change of variable, the form 

 alone of the results being material. The last equation is a tri- 

 nomial, wherein 



L =hf+ha—d 2 , 



]X=ty+ke—e% 



M = hg + kf—(n— \)he+ [n + \)lca— 2ae, 



§+3«f +3^=0, (e) 



and rj by the relation 



i{iog(r,)}=-3«, co 



and C by 



&{=-c (g) 



When £ = »7=£, the terordinal is soluble as a primordinal. When (e) is 

 insoluble, a transformation of (d) may possibly have a ccesura which is 

 soluble. 



