ON THE RECENT PROGRESS OF ANALYSIS. 291 



22. In the twelfth volume of Crelle's Journal (p. 173), Dr 

 GuetzlafF has investigated the modular equation of transforma- 

 tions of the seventh order : it is, as we know from the general 

 theory, of the eighth degree, and presents itself in a very re- 

 markable form, which closely resembles that in which M. Jacobi, 

 at p. 68 of the Fundamenta Nova, has put the modular equa- 

 tion for the third order. Dr Sohncke has given, at p. 178 of 

 the same volume, modular equations of the eleventh, thirteenth 

 and seventeenth orders, none of which apparently can be reduced 

 to so elegant a form as those^pf the third and seventh. Possibly 

 the transformation of the thirty-first order might admit of a 

 corresponding reduction. The whole subject of modular equa- 

 tions is full of interest. Dr Sohncke has demonstrated his re- 

 sults in a subsequent volume of the Journal (xvi. 97). 



In the fourteenth volume of Crelle's Journal there is a paper 

 by Dr Gudermann on methods of calculating and reducing in- 

 tegrals of the third kind. I have already quoted from this 

 paper the expression of the opinion of its learned author, that it 

 is impossible to express the value of integrals of the circular 

 species in terms of functions of two arguments. If this be so, 

 it is impossible to tabulate such integrals, and therefore our 

 course is to devise series more or less convenient for determining 

 their values when any problem, e.g. that of the motion of a 

 rigid body, to which Dr Gudermann especially refers, requires us 

 to do so. The formation of such series is accordingly the aim 

 of this memoir, which contains some remarkably elegant formulae; 

 one of which connects three integrals of the third kind with 

 three of the second. 



In the sixteenth and seventeenth volumes of the same Jour- 

 nal, Dr Gudermann has given some series for the development 

 of elliptic integrals; and he has since published in the same 

 Journal a systematic treatise on the theory of modular functions 

 and modular integrals, these designations being used to denote 

 the transcendents more generally called elliptic. The point of 

 view from which he considers the subject has been already in- 

 dicated (vide supra, p. 271). In a systematic treatise there is of 

 course a great deal that does not profess to be original, and it is 

 not always easy to discover the portions which are so. Dr 

 Gudermann's earlier researches are embodied and developed in his 

 larger work ; and in some of the latter chapters (xxiu. 329, &c.) 



192 



