On the Theory of Equal Roots and Multiple Points. 375 

 duced and assume the more perfect or prismatic forms ; but when 

 excess of ammonia is present, the formation of the crystals is hur- 

 ried, and the stellate feathery crystals result. That the feathery 

 crystals are truly mere aggregations of theprisms is shown by he 

 fact, that where they are more slowly formed from a very dilute 

 solution, we occasionally meet with the prisms projecting from the 

 sides of the feathery portions. This explanation of the respective 

 conditions under which these two forms are met with w apph 

 cable to their occurrence naturally m animal fluids When 

 urine is kept, it gradually becomes ammomacal; and as the 

 ammonia is formed*, it combines at once with the phosphate of 

 magnesia; hence the prisms are produced ; and we never find 

 the feathery forms in either the urine or other animal fluids, 

 unless some portion of the liquid has been prevented from ad- 

 mixture witlAhe remainder by mucus, or some « her such sub- 

 stance until after the fluid has become ammomacal. ihe pen- 

 niform" crystals of the ammonio-magnesian phosphate are merely 

 modified films of the stellate; they may be «^ ° b ^*l by 

 adding excess of ammonia to a very dilute solution of the phos- 

 phate of ammonia and sulphate of magnesia, and are remarkable 

 for the curvatures of then- rays. 



9 St. John's Square, 

 March 1, 1852. 



LIV On a remarkable Theorem in the Theory of Equal Roots and 



Multiple Points. By J. J. Sylvester, Bamster-at-Law*. 

 TN order that the theorem which I propose to state may be 

 F the more easily understood and with the least ambigu^ 

 expressed, I shall commence with the case of a homogeneous 

 function of two variables only, x and y. 

 Let , 



= aod n + nbx n- Ky + n. %£- cx n -* ■ y* + &c 



+ . . . +nb'xy n - l + a'y n , 

 and let the result of operating with the symbol 



P L - c „ a ,. h' a' be called the Evectant of such 

 Sa^fci^-Wi «i process r tunes the 



^ nu^tand by the multiplicity of the equation the number of 



equalities betwee, the roots that exist; so that a pair of equal 



* Communicated by the Author. 



