ANALYTICAL DEMONSTRATIONS 

 OF DR MATTHEW STEWART'S THEOREMS*. 



IN 1746 Dr Matthew Stewart, the father of Dugald Stewart, 

 published his "General Theorems." He was at that time a 

 candidate for the chair of mathematics at Edinburgh, then vacant 

 by the death of Maclaurin ; and his success is attributed to the 

 celebrity which these remarkable propositions immediately ac- 

 quired. They were enunciated by Dr Stewart without demon- 

 strations, and remained undemonstrated till 1805. Mr Glenie, 

 in the Edinburgh Transactions for that year, has given a geo- 

 metrical method by which the General Theorems and other 

 similar results may be established. 



But as yet they have not, I believe, been proved, except by 

 Geometry; and in an article in the 17th volume of the Edinburgh 

 Review, ascribed to Playfair, they are strongly recommended to 

 the attention of analysts. It is hoped, therefore, that the follow- 

 ing attempt will have some degree of interest. 



We shall begin by establishing a general proposition, from 

 which all the theorems in question, and many others, may be 

 deduced. 



LEMMA. If f<f> is a rational and integral function of sin^> 

 and cos <, then a value may always be assigned to n, such that 



shall be independent of <. 



The preceding expression is equivalent to 



(1 . + D + . . . D n ~ 1 } /</>, (where D$ = (/> + ^ 



* Cambridge Mathematical Journal, No. XII. Vol. n. p. 371, May, 1841. 



