Asymmetrical Probability-Curve. 95 



equations ; the whole system being 



dy _ 1 c% 

 S& " 2 eta 2 ' 



(2) 

 (3) 



d/ 6 c?^ 3 ' 



To these data are to be added the conditions (a) that^'-rA^ is 

 small*, and (b) that 



■+oo 



dyx = lf. 



. 



Of the problem thus stated the following two solutions are 

 offered : — 



First Method. — From equation (1) we have by a familiar 

 method 



*=MvrJ> :■ (4) 



where <j> is an arbitrary function. 



From equation (2) by a known method J we have 



+(' + -f-'( 8 a) + -)* ; - • (5) 



where ^ and yfr 2 are arbitrary functions of k (and^'). These 

 functions are restricted by equation (4) to the form 



hnil 



To further determine the forms of i/r, and yfr 2 we must utilize 

 equation (3); from which, by combination with equation (2), 

 we have 



dy__l d % y 



dj~~ Sdxdk {) 



This condition may be fulfilled by assuming yfr 1 and likewise 



\jr 2 to consist of a series of ascending powers of ^ ; which 

 is permissible by condition (a). * 



* As shown above, p. 92 note. t A condition of a probability-curve, 

 t Forsyth's ' Differential Equations/ Art. 256. 



