FR()^i i:lemi:ntar\' algebra to the calculus. 417 



x'-\-y'-^d- giv^es xdx-\'ydy ^0, because "division by" dx 

 gives the proper Leibnitz form; hut udx -{- bdy = c is mean- 

 ingless. 



(ii) One often wonders what ' expansion ' in power series means 

 psychologically — why sliould we be so anxious always to 

 expand in powers of x ? We can answer the question in 

 the case of our ordinary decimal notation : the process is an 

 extremely convenient way of counting with 10 (or rather 9) 

 figures and recording each complete cycle. 



But the more general process involved in Maclaurin's 

 Theorem has a more elusive vaison d'etre. Is it really, as it 

 seems to us, logically inevitable ? Could mathematical 

 reasoning have developed satisfactorily on other lines and 

 missed ' expansion ' ? The only semi-answer that I can 

 give myself to these questions is that expansion is a form of 

 integration by parts, which is apparently an inevitable 

 sequel of integration itself ; and integration is a fundamental 

 logical process, inevitable to the human mind. 



South African Homoptera — of all the orders of 

 insects in South Africa, the Hemiptera, and i)articularly the sub- 

 order Homoptera. have been studied the least, and the list of 

 described species would scarcely numl)er more than one hundred. 

 Mr. E. S. Cogan, M.A., has recently contributed to our know- 

 ledge of South African I-Iomo])tera the results of his study of a 

 series of South African Cercopida' and Jassoidea, hitherto 

 scarcely known at all, whicli had been sent to Ohio State Univer- 

 sitv by Mr. C. W. Mally, of the Department of Agriculture, 

 Capetown. In all some 38 forms were studied, and the results 

 have now been published.* In the course of his descriptions the 

 author observes that the practice of burning the veld, though 

 not very strongly recommended by botanists, nevertheless serves 

 to keep down the grass-feeding species of Jassids. The pro- 

 tective resemblance to plants and flowers borne by many African 

 Homoptera are specially mentioned. 



''Ohio Joiinial of Science (1916) 16 [5] 161-200. 

 B 



