472 



which preceded his departure from England, he studied with un- 

 remitting ardour and perseverance for no less than eighteen hours 

 each day ; and, conquering every difficulty, acquired a competent 

 knowledge of the principles of science, learned the uses of various 

 instruments, and made himself thoroughly master of the methods of 

 taking observations both at sea and on land. 



The narrative proceeds to notice the arrival of Mr. Douglas in 

 America, the progress of his undertaking, the loss of his collections 

 and most of his books and papers, by the upsetting and dashing to 

 pieces of the canoe in which he attempted to pass the rapids, and, 

 lastly, his death in 1833, at Owhyhee, in the Sandwich Islands, 

 whither he had proceeded on his return to Europe. 



The books which were preserved, and which have been received 

 by Major Sabine, consist of several volumes of Lunar, Chrono- 

 metrical, Magnetical, Meteorological and Geographical observa- 

 tions, together with a volume of field sketches. The geographical 

 observations of latitude and longitude refer to two distinct tracts 

 of country ■ first, the Columbia river, and its tributaries ; and the 

 district to the westward of them : and, secondly, California. Mr. 

 Douglas very judiciously selected the junctions of rivers, and other 

 well characterized natural points, as stations for geographical de- 

 termination. The papers containing the details of his magnetical 

 inquiries comprise records of observation of the dip, and of the in- 

 tensity, at various stations both in North America and in the Sand- 

 wich Islands. 



"Analysis of the Roots of Equations." By the Rev. R. Murphy, 

 M.A. Communicated by John William Lubbock, Esq., F.R.S. 



The object of this memoir is to show how the constituent parts 

 of the roots of algebraic equations may be determined by consi- 

 dering the conditions under which they vanish; and, conversely, to 

 show the signification of each such constituent part. 



The following are the propositions on which the author's investi- 

 gations are founded. 



1. In equations of degrees higher than the second, the same con- 

 stituent part of the root is found in several places, governed by the 

 same radical sign, but affected with the different corresponding 

 roots of unity as multipliers. 



2. The root of every equation, of which the coefficients are ra- 

 tional, contains a rational part ; for the sum of the roots could not 

 otherwise be rational. This rational part, as such, is insusceptible 

 of change in the different roots of the same equation ; consequently 

 its value is the coefficient of the second term, with a changed sign, 

 divided by the number of roots, or index of the first term. 



3. The supposed evanescence of any of the other constituent 

 parts, implies that a relation exists between the roots ; and if such 

 a relation be expressed by equating a function of the roots to zero, 

 that constituent part will be the product of all such functions, and 

 a numerical factor. 



4. The joint evanescence of various constituent parts, implies the 

 coexistence of various relations between the roots, and that an in- 



