July 12, 1894] 



NA TURE 



263 



the order of the eliminant will be the number of common points 

 of the surfaces, less the number of common points which lie in 

 the two planes c = o, w = o. In the paper several examples are 

 considered. It will suffice to mention here the case which led 

 to these investigations. Consider two equations of the form 



a-jX-! 



»3+ 9 



oi + e oj + 9 

 These equations may be written in the homogeneous form 



a;i(/ + 9 031^ + 9 ' 



a,i/i + 9 



and thus represent curves of the third order. But the curves 

 have evidently three common points on the line 1^ = 0; and a 

 common node (with different tangents) at the point 9 = o, i(/ = o. 

 Hence the degree of the eliminant of the given equations is 



9-3-4 = 2. 

 More generally, the eliminant of two equations of the form 

 "« + »„_i + .. + u,^ = o, 



■^m + C',,,., + ... + Vf, = O, 



where lu and vi are homogeneous expressions of the nh degree 



"i + 9 



, + 9 



+ 9' 



is shown to be of degree (r - i)mn in the variables jt,, .x.,, ... Xr. 

 — Solvable cases of ttie motion of a top or gyrostat, by Prof. A.-G. 

 Greenhill, F. R.S. When the sum of the parameters of the two 

 elliptic integrals of the third kind, whose poles correspond to the 

 highest and lowest position of the axis of the top, is an aliquot 

 part of the associated elliptic function periods of the form 

 K + f'K'i, where /is a proper fraction, then Abel's theory of 

 Pseudo- Elliptic Integrals can be utilised to construct an alge- 

 braical S'llution of the motion, provided that in the general case 

 a secular term // is associated with the azimuth i^ of the axis. 

 Denoting by 9 the angle between the axis of the top and its 

 highest position, and supposing 9 to oscillate between a and S, 

 a>8>0, then the dynamical equations to be satisfied are of the 

 form 



'M3 



+ iA sin- ( 



Asin^9'^ 

 lit 



r Cr cos 9 = 0; 



cos 9), 



equivalent to, with «- = Wgi/A, 

 sin B = n ^2 .^'(cosh 7 — cos 9 , 



lie 



cos 3 - cos 9 

 C''cos 9 



cos 9 —cos o), 



■sJjzKVlgh) 



and 



v'(cosh 7 - cos 9 . cos 5 - cos 9 . cos 9 - cos o) ' 



it is found that the constants in the problem can be 

 expressed in terms of two arbitrary constants m and c. 



Jn the simplest case, / = A, the motion of the axis, or of a 

 fixed point on it, will be given by the equation 

 sin -Be-^^-f"' = (cos 9 - IJ) ,y(cosh 7 - cos 9 . cos 3 - cos 9) 



-I- /(E cos 9 - F) V(cos 9 - cos o), 

 where 



cos /3 = : 



VM 

 - I - ( 



cosh 7 = 



^/M 



VM 

 M = (/«-- I -(- c -f c-f + 8{m + I) (i- 



+ C-). 



D = 



3OT- + 4m + I - C 



F = : 



£ _ {2m + 1)^J2 



' + 3m- + 2{l + c + c-)m -t- I .^ 3<r -^ 3<-2 



where 



p _ 2m + I G- _ 2;k' C-r2 



n 2 ^M • 2AW^ ~ ;^m' 2AWp/ 

 t ifi - 2(1 - a)m^ + (i - 6g + a-)/;;" -4(a - a-)//; -H 4a- 



a = <■ + c". 

 NO. 1289, VOL. 50] 



The rnodulus /■ of the associated elliptic functions is given by 

 k = c/{i + <:), so that the period of the axis between its highest 



and lowest position is —^-"f- times the period when the 



(I + c)^2 

 body makes plane oscillations, swinging through an angle 

 4 sin -V/(i + i:). 



By putting m = - i, the secular term // is made to dis- 

 appear, and the path of a point on the axis of the top is a 

 closed curve with four branches ; this curve has four cusps if 

 c = h, and it has four loops if c > i. 



The term Cr can be made to disappear by determining a in 

 terms of m by the solution of a quadratic equation, and the 

 motion is that of a spherical pendulum ; but it is not possible 

 now to make p vanish. In the next case of/ = J the motion of 

 the axis is given by an equation of the form 

 sin SflfSij.-//)! = 



(cos -9 - D cos 9 -(- D') ^(cosh 7 - cos 9 . cos 9 - cos a). 



+ i{E cos -B - F cos 9 -^ F') ^'(cos - cos 9) 



and 



C0SV3 



I + 4c - S^- + + c* 

 J + 4r- 5^ « + 4c^ - c* 



VM 



cosh 7 = "'' +'+ ° - 5'" + 4^ - '* 

 VM • 



M = {m- - I + 8<- _ 15^-2 + 8£.3 _ ^4)2 



-f 8(1 -f)2(2r-^S)(, _ 2c){m+ I - 



^,2 _ 2 c3 - C* p_ 3 W -f- (I - 20(2 



(I -^"(i +c)' 



E = 3 '" + ( ^ ~ ^'^ (' 



VM 



3^ + c"-), 

 c) 







34'M 

 V2.. 



]J, D', E, E' being readily found by verification. In this case 

 the secular term // is destroyed by taking m= - i(i -2c) {2-c), 

 and a point in the axis now describes a curve with six loops'. 

 The case of/ = 3 can similarly be made to give a curve with 

 three loops, the general state of motion being given by an 

 equation of the form 



sin'9f3(+-/')' = (cos=9 - D cos 9 + D') 



v'(cos;3 -cos9 . cose - cos a) 

 + i(Z coi- 9 - F cos 9 -f F') V(co3h 7 - cos 9). 

 A similar procedure will serve for 



« 3 I. 2, 3. 4 

 4 



f- 



&c. 



4 S 



the results are of rapidly increasing complexity, but the con- 

 stants D, E, ... are readily determined when/ is known, E 

 being the simple multiple ^^^'2 of//«, while 



P _ M"' + p 



« n i'M ' 



where A fpiPu-Fv) + ^pV 



" / P» - 



^</« 



is the pseudo-elliptic integral corresponding to a parameter v^ 

 which is a ^th part of a period. The Rev. F. J. Smith, F.R.S.^ 

 of Oxford, has constructed an apparatus by which the preceding 

 theory can be tested, and the agreement between the predicted 

 and experimental results is very satisfactory.— Impromptu com- 

 munications were made by Dr. J. Larmor, F. R.S. (on the wave 

 surface), and Dr. M. J. M. Hill, F.R.S. (on Monge's solution 

 of a differential equation).— .\t the special meeting, held on the 

 same evening, for considering certain resolutions relating to the 

 incorporation of the Society under the Companies .\ci 1867, 

 authority was given to the Council to carry out the incor- 

 poration. 



Paris. 

 Academy of Sciences, July 2.— M. Lcewy in the chair.— 

 Researches on plienylhydrazine. Action of oxygen and of 

 water ; formation of salts, by M. Berthelot. Oxygen reacts on 

 a solution of phenylhydrazine hydrochloride, giving off a volume 

 of nitrogen equal to that of the oxygen absorbed," and yielding 

 an uncrystallisable oily compound answering to the reactions of 

 diphenylhydrazine. Pure anhydrous phenylhydrazine heated at 

 100' with oxygen in sealed vessels yields about half as much 



