On Homogeneous Functions, and their Index Symbol. 129 



small things with great, the cyclones of Colonel Reid. To pre- 

 serve equability of temperature, a screen of pasteboard was often 

 found serviceable. Now that a needle suspended from a silken 

 fibre sixteen inches long, covered with a glass shade and placed 

 in strong sunlight, which are the conditions of Dr. Goodman's 

 experiments, should also be influenced by air-currents, is exceed- 

 ingly probable, and that a permanent current of electricity should 

 circulate in a helix of covered copper wire with its two ends dis- 

 connected being exceedingly improbable, it appears worth the 

 trouble to subject the matter to one or more of the three testa 

 which have been above proposed. 



Queenwood College, 

 Jan. 1, 1852. 



XX. Homogeneous Functions, and their Index Symbol*. 

 By Robert Carmichael, A.B., Trinity College, Dublin-^. 



IN a valuable memoir published in the Philosophical Transac- 

 tions for the year 1 844, Professor Boole, by the aid of two 

 fundamental principles, has given general methods of solution 

 for certain extensive classes of linear differential equations. It 

 is the chief object of the present paper to show that, by a gene- 

 ralization of those principles and a suitable development of the 

 consequences of the higher principles, we can obtain similar 

 general methods of solution of corresponding classes of -partial 

 differential equations. The solutions of such partial differential 

 equations will be found to be unaffected by the number of inde- 

 pendent variables whieh the equations may contain ; but more 

 especial reference is made to those in most common occurrence, 

 containing but two independent variables, x and y. 



In the course of the investigation, extensions of many familiar 

 and elementary theorems are furnished, which seem to possess 

 much practical utility. From the spontaneity with which they 

 evolve themselves, and the facility with which they admit of. 

 employment, they appear to open a large and fruitful field for 

 future speculation. 



By an application of the general principles to the subject of 



* The principal portion of the first seven articles in this paper has been 

 already published in the Cambridge and Dublin Mathematical Journal, 

 November 1851. In the Number of the same Journal for February 1851, 

 will be found an interesting and masterly paper by Mr. Sylvester, "On 

 certain general Properties of Homogeneous Functions." The same symbol 

 is there applied to equations in finite terms, with a view to the subjects of 

 surfaces and the linear solution of systems of indeterminate equations. In 

 the present paper the symbol is applied to the subjects of partial differential 

 equations and multiple definite integrals. 



t Communicated b\ the Author. 

 Vhd. Mug. S. 4. Vol. 3. No. 10. Feb. 1852. K 



