440 Sir James Cockle on a Differential Criticoid. 



which it gives upon the screen, with its graduated circle and 

 needle, the great range of delicacy which may be given to the 

 instrument by varying the coil and needles, so that all ex- 

 perimental requirements may be answered, and, finally, the 

 satisfactory character of its performance as a demonstration 

 galvanometer, all combine to justify the record which is here 

 made of it. 



Philadelphia, April, 1875. 



LIII. On a Differential Criticoid. By Sir James Cockle, 

 F. R.S. } Corresponding Member of the Literary and Philoso- 

 phical Society of Manchester, President of the Queensland Phi- 

 losophical Society, Sfc* 



1. "OESIDES the criticoids discussed in this Journal, at 

 -D places to which my paper " On Primary Forms " in 

 the Number for February last (1875) will give means of refer- 

 ence, there is another, which I call a differential criticoid. It 

 indicates a certain relation between two differential equations, 

 quantoids, or quotoids, whereof one is a transformation of the 

 other by change of the independent variable f. 



* Communicated by the Rev. Robert Harley, F.R.S. 

 f The synthetic solutions, which I gave in art. 6 of my paper "On Primary 

 Forms," and the footnote thereto, admit of convenient developments. Let 



h denote the operation U U, with this restriction^that no arbitrary con- 

 stants are to be added. And let k denote \ r)\ £ with the same restriction. 

 Then 



pM^- = ^1+0^^1, 



the operands on the dexter being unity. By developing the fractions in 

 the operators we obtain two series. And by a similar process we should 

 obtain the synthetic solution of a terordinal as an aggregate of three analo- 



gous series. If, extending the notation of that article, we put — - =r, then 



da 2 

 the most general equation reducible to one with constant coefficients by a 

 change of the independent variable is 



3-(j + «*)**-0 M 



Here if we take £ = e 2az p and rj=—e— 2az p, the form of \]/{cc) is suggested ; 

 for when h\p(x)=-^(x), then yf/(x) is a particular integral. *issume 

 \]/(x) = e mz ; then, if we add no arbitrary constants, 



r* „ C r* „ e {m-2a)z e mz 



\ e***p \ -e-*<*zpe mz = C e 2az p — = —1 x (b) 



J* J J* 2a- m m(2a-m) v y 



' m{2a-m) = \, or (m — a) 2 = a 2 — 1, then e^will be a particular 

 and the complete integral will be 



y = Ce(°+ ^ 2 -l>+C 2 e(a- VaTT^ ( c ) 



