November 30, 1853.) 



SCIENCE. 



717 



process of taking place. The present writer's 

 own image with .Shelley's lines above quoted 

 is not so much of dead leaves actualh' moving, 

 as of the leaves rustling, with the sense or 

 feeling that thej- are driven bj- the wind. The 

 words descriptive of motion give, rather, the 

 feeling of action connected with the leaves, 

 than a picture of movement itself. So, to say 

 that the mountains rise is to direct the mental 

 e\-e upwards, rather than to introduce any pic- 

 ture of objective motion into the mental land- 

 scape. So, then, it seems probable, that, while 

 we notice moving rather than resting things, 

 our mental pictures tend to be representations 

 of resting attitudes, rather than pictures of 

 motion. And the greater vividness which de- 

 scriptions of motion nevertheless possess would 

 seem to be due to the sense of activity that 

 the^- introduce into our ideas of the objects ; 

 and that this sense is connected with the mus- 

 cular sensations that we are accustomed to 

 associate with all clearly perceived motions 

 seems both probable in itself, and in some wise 

 confirmed bj- Professor Strieker's observations. 

 The whole leads us, in fact, to another probable 



law of mental life ; viz., that, since an animal's 

 consciousness is especially useful as a means 

 of directing his actions, the ideas of actions, 

 however tlwy are formed, will naturally be 

 among the most prominent elements of the 

 developed and definite consciousness. We 

 need not make any assertion about the direct 

 source of these ideas. AVhether the active 

 muscular sense is a direct consciousness of 

 the outgoing current, or a true sense through the 

 mediation of sensory nerves, the result will 

 not aflTeet either Professor Strieker's argument 

 or our own suggestions. 



In conclusion it may be well to saj-, that, if 

 psychology were already a developed experi- 

 mental science, such independent and hasty 

 observations and generalizations as our au- 

 thor's would hardly be worth discussion. But 

 as things are, even very imperfectly conducted 

 observations, if thev are direct and sincere, 

 must be thankfully accepted. Something of 

 the same sort may possibl}- hold good of the 

 similarlj- hasty suggestions that have here been 

 thrown together. 



JOSIAH ROYCE. 



WEEKLY SUMMARY OF THE PROGRESS OF SCIENCE. 



MATHEMATICS. 

 Algebraical equations. — M. Walecki, in a note 

 presented to the Acaileraie des sciences by JI. Her- 

 mite, gives a proof of a fundamental theorem in the 

 theory of algebraical equ.ations; viz., that every alge- 

 braical equation h.is a root. The theorem being evi- 

 dent for real coefBcients, M. Walecki assumes the 

 coefficients as imagin.iry, and writes the first mem- 

 ber of the equation in the form P -|- iQ, and also 

 makes F(x) — P^ + Q-. He considers first the case 

 of an equation of odd degree, say p ; then it is only 

 necessary to prove that the equation F{x) = 0, of de- 

 gree 2 p. has a root. To do this, he writes x = y + z, 

 and distinguishes the odd part in z from the even 

 part in the development of F{y + z), writing thus: 

 F{x) = <?{z-) +zil'{z-). The resultant of ?> and ijj is 

 shown to be a real polynomial of odd degree in y, and 

 vanishing for a real value of y. Two cases present 

 themselves: viz., one of the functions <p or V niay 

 vanish identically; and this can only be V> for the 

 coefficient of the term of highest degree in <!> is not 

 zero. Then, being of odd order, F{x} has a real 

 divisor of the second degree. Tlie second case is 

 when i> is not identically zero, and when <p and ii 

 have a common divisor, F(x) being then decomposed 

 into the product of two factors. The author shows, 

 then, that in eitlier case a divisor of F{x) is obtained 

 of either the first or second degree, and with real co- 

 efficients ; thus proving the proposition for an equa- 

 tion of odd order. A similar investigation is given 



for equations of even order. — ( Comptes rendus, March 

 10.) T. c. [409 



A differential equation. — M. I'abbg Aoust has 

 here given a method for obtaining the formula giving 

 the general integral of the differential equation — 

 d"ij . d"-'-y 



^'•dx" + ^'»^"-'d^"-l + ---+^ny = Fix), 



by aid of a certain multiple definite integral. The 

 quantities Ai, A^ - . ■ An are constants. He pro- 

 poses first to solve the problem of finding a function, 

 <p, in terms of another function, V' ; the two functions 

 being connected by the relation — 



i,(x) 



J^danfdan-i...j^da,9[a„''-...a,'''xj- 

 The process for the reduction of this is by substituting 



successively z, for a,"'!, z^ for Oo^'z, etc.; and 

 finally the expression of 6 in terms of V is obtained. 

 The transition from the solution of this problem 

 to the solution of the problem of finding'the general 

 integral of the given differential equation is then in- 

 dicated, and the integral given in the form — 



n „i 1 /"' r' 



y = 'S Mix" + —~ tdanldon.i. . . 



rF{a„a„-i . . . a,i)do,. 



•'0 



The quantities J/,, J/j . . . Af„ are arbitrary con- 

 stants, and a,, etc., roots of a certain algebraical 

 equation. — {Comptes rendus, iinrch 19.) t. c. [410 



