with Mr. T. S. Davies's Notes on some of the Topics. 359 

 transformation is inadequate to the removal of more than one 



term at a time. . r j ■ j 



To Tschirnhausen is due the introduction ot quadratic and 

 the sugtreslion of higher transformations. Without this im- 

 provement all progress in the theory of transformation would 

 have been stopped. The two different quadratic transforma- 

 tions which he gives for the annihilation of the middle terms 

 of a cubic are well- worthy of attention. In the first of them 

 (Acta for 1G83, p. 206), the equation which connects the roots 

 ofthe original and transformed equations involves first and 

 second po"wers of the former roots, and first powers only of 

 the latter; but in the second transformation ot a cubic (Ibid. 

 p. 207) the case is reversed, and the original roots only enter 

 to one dimension, while the roots of the transformed equation 

 enter to two. To Tschirnhausen must be given a very ex- 

 alted position among the cultivators of the theory of equations. 

 Elevation of degree from elimination would seem' to impose 

 insuperable obstacles to any material extension of Tschirn- 

 hausen's views. However, by a combination of the indeter- 

 minate method with that of Tschirnhausen, Mr. Jerrard has 

 overcome this difficulty ; and as one of his earliest results, I 

 may mention his transformation of the general equation of the 

 fifth deoree to a trinomial form. Mr. Jerrard has shown how 

 to annihilate any three of its four middle terms. It is true 

 that in the transformations of Tschirnhausen the transformed 

 equations are indeterminate ; but then no more indeterminate 

 quantities are introduced than there are conditions to satisfy. 

 The transformed equations of Mr. Jerrard are indeterminate 

 in another sense ; they involve more disposable quantities than 

 the number of conditions to be satisfied by the coefficients of 

 the transformed equation. The difference between Tschirn- 

 hausen's and the modern method of determining the roots of 

 the original cubic, when those of the transformed are known, 

 will be°seen on comparing the course pursued at p. 206 of the 

 Acta (1683) with that given at pp. 27-29 of Mr. Jerrard's Re- 

 searches, and subsequently extended (Ibid. pp. 36-39). As to 

 the eliminations requisite for the formation of the transformed 

 equation, they may either be performed directly, or may be 

 made to depend upon the theory of symmetric functions; and 

 in the latter case, a knowledge of the number of the roots ot 

 a given eciuation is taken for granted. In effecting the trans- 

 formation above alluded to of the equation of thehtih degree, 

 the principal difficulty consists in forming the new equation; 

 that once done, the difficulty of giving the required form to 

 its coefficients is not great. In its application to this trans- 

 formation, my method (of vanishing groups) only requires us 



