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LXV. Notices respecting New Books. 

 Finite Integrals and the Doctrine of Germs, an Essay by S. Earn- 

 shaw, M.A. Cambridge : printed at the University Press, 1876. 

 (Pp. 32.) 



^HE integration of the equation F( j-, -j-, -r-i • • • J u = 0, the 



general linear equation with constant coefficients, is the sub- 

 ject of the present essay. It is well known that the equation is 

 satisfied by the function 



u=Ce lx + m y+ nz •■■ , 



I, m, n . . . being arbitrary constants, provided 



Eft m, n ...)==0. 



The author maintains that this is the general integral, provided 



Z, m, n, ... are germs. And a germ is a sort of glorified arbitrary 



constant, which he defines to be " an essentially indefinite quantity." 



But our author's meaning will come out more clearly if we take a 



d 2 u _ du 

 particular case. Let us then consider the equation -y-^ j-. 



From what has gone before, it follows that the " germinal" solution 



of this equation is u =Ce mx + m y (1) 



This may be written in the form 



w = 1+ f- m+ (o + f) m2+ (ri3 + o) m3+ -- 



The coefficient of each of the powers of m satisfies the equation 

 and is a sub-integral. And the expanded form of u is the general 

 solution, provided m, m 2 , m 3 ... be considered as independent con- 

 stants ; i. e. the general integral is 



*= C + C .-?+ C *(A + !)+ C <T^ + I^) 



+. .. 



"With this understanding as to the meaning of m, (1) represents the 

 general integral in a finite form. 



Germs, according to our author, are of two kinds : — minor germs, 

 which are simple constants added to the variables ; major germs, 

 which enter the function in any other way. Sometimes these 

 germs are latent. Thus the minor germs are latent in (1); for if 

 cc-\-g and y + h were written for x and y, it would merely introduce 

 two factors which would coalesce with the arbitrary constant. It 

 is, however, possible to put the general integral into another form, 

 in which the major germs shall become latent and only the minor 

 germs appear explicitly. This form in the case of the above equa- 

 tion is p+g-)2 



u = C(y + 7i)-*e~ 4(i,+h). (2) 



the expressions (1) and (2) being indifferently the general integrals 

 required. 



In some cases, as might be expected, an expression containing a 

 germ may be replaced by an arbitrary function of the variables : 



