344 



NATURE 



\Augnst 23, 1877 



In regular Irochoidal waves the particles move in vertical 

 circles with a constant velocity and are always subject to the 

 same pressure. Of the energy of disturbance half goes to give 

 motion to the particles and half to rai^e them from their initial 

 position to the mean height which they occupy during the 

 pas'irige of the wave. 



Now the mean horizontal positions of the particles remain 

 unaltered by the waves, hence, since their velocities are constant, 

 none of their energy of motion is transmitted ; nor since the 

 pressure on each particle is constant can any energy be transmitted 

 by pressure. The only energy therefore which remains to be trans- 

 mitted is the energy due to elevation, and that this is transmitted 

 is oljvious since the partirles are movin,' forward when above 

 their mean position, and backward when below it. This energy 

 constitutes half the energy of disturbance, and this is therefore 

 the amount transmitted. 



For a defiiiie mathematical nroof that — 



In wives on deep luater the rate at which the energy is cairied 

 forward is 4 the energy of disturbance per unit of length X ''')' the 

 rate of propagation. 



I,et //„ be the initial height occupied by a particle supposed to 

 be of unit weight, h-^ the height of the centre of the circle in 

 which it moves as the wave passes, r the radius of the orbit, and 

 e the angle the radius vector makes with the horizontal diameter, 

 then the height of the particle above its initial position is 

 //[ -/;(, + )■ sin fl, adding to this the height due to its velocity 

 we have the whole energy of disturbance — 

 = 2(/;i - h^ + rsine. 

 The velocity of the particle i s — 



= -/2^(//, -/;„), 

 end the horizontal component of this is — 



= '•^zfiyh^ - //^.sin 9. 

 Therefore the rale at which energy is bemg transmitted by the 

 particle — 



= {2 (/', - /'„) + rsin e] Vi7(//i - '''0") • sin 6. 

 and the mean of this — 



^/i 



2 ('', - /'„) + r sin e| \'2g (//j - //„) sin i d 9 



= 3'- '^2^»-(/;i-l„), 

 and if \ be the length of the wave and n \ the rate of propa- 

 gation — 



7 7 V r^ -i 2 c A 



h, - hi, = ana — ^^ = Awr', 



\ \ 



. ■. the mean rate at which energy is transmitted by this particle 



= n\ (y^j - //„), 



or the rate of propagation multiplied by half the energy of dis- 

 turbance. Q.E.D. 



It now remains to come back to the speed of the groups of 

 waves and to show that if the rate at -which energy is transinittid 

 is equal to the rate of propagation multiplied by half the energy of 

 aistnrbance, then the velocity of a group of waves will be \ that of 

 the indiv'dual wtrr'es. 



Let /'i, P^, F.^, P^ be points similarly situated in a series of 

 waves which gradually diminish in size and energy of disturbance 

 from Pi, to /"i, in which direction they are moving. Let E be 

 the energy of disturbance between j"j and P^ at time t, E + a 

 the energy between P2 and /j, E + 2a between P.^ and P^, and 

 so on. 



Then at the time t + n after the wave has moved through one 



wave-length it follows that the energy between P^ and P, 



will be — f 



_ E + E + a 



= E + 



and between P^ and P.^ wUl 



_ E + a + E + 2a 



E + 



3« 



and again after another interval, u, the energies between P^ and 

 P„, P^ and /"a will be respectively — 

 E + ^+ E + lJh 

 = I L = E + a, 



S« 



So that after the waves have advanced through two wave-lengths 

 the distribution of the energy will have advanced one, or the 

 speed of the groups is .^ that of the waves. Q.E.D. 



Of course this reasoning applies equally to the waves on the 

 suspenr^ed balls, when connected by an elastic string, as to 

 water ; and in this c.Tse the conclusions may be verified for, as 

 on water, the groups of waves travel at a .slower rate than the 

 waves. This experiment tends to throw light on the manner in 

 which the result is brought about. When a ball is disturbed, 

 the disturbance is partly communicated to the adjacent ball by 

 the connecting string, and part retained in the form of pendulous 

 oscillation ; that part which is propagated forward is constantly 

 reduced in imparting oscillations to the successive balls and soon 

 dies out, while the motion retained by the swinging pendulum 

 con^tantly gives rise to succeeding waves until it is all absorbed. 

 If the tightness of the cord be adjusted to the length of the 

 suspending threads, waves may be made to travel along in a 

 manner closely resembling the way in which they travel on 

 water, the speed of the group being \ the speed of the individual 

 waves. 



Although the progression of a group has hitherto been spoken 

 of as if the form of the group was unaltered, this is by no means 

 the case as a rule. 



In the mathematical investigation it was assumed that the 

 motion of the particles is circular ; this, however, cannot be the 

 case when the succeeding waves differ in size by a sensible quan- 

 tity, and hence in this case the form of the group cannot be 

 permanent. .\nd it may be further shown that as a small group 

 proceeds, the number of waves which compose it will continually 

 increase, until the gradation becomes indefinitely small ; and 

 this is exactly what is observed, whether on water or on the 

 strings. 



.So far as we have considered deep water, when the water 

 is shallow compared with the length of the waves, the results 

 are modified, but in this case the results as observed are strictly 

 in accordance with the theory. 



According to thi-, as waves enter shallow water the motion of 

 the particles becomes elliptical, the eccentricity depending on 

 the shallowness of the water ; and it may be shown that under 

 these circumstances the r.ate at which energy is transmitted is 

 increased, until when the elliptic paths approach to straight 

 lines the whule energy is transmitted, and consequently it follows 

 that the rates of the speed of the groups to the speed of the waves 

 will increase as thewater becomes shallower, until theyare sensibly 

 the same. In which case only the groups of waves are perma- 

 nent, and Mr. Scott Russell's solitary wave is possible, liesides 

 the expl mat ion thus given of these various phenomena, it appears 

 that we hive here a means of making some important verifica- 

 tions . f: he assumptions on which the wave theory is based ; 

 for the rcUiive speed of the groups and the waves which com- 

 pose them affords a criterion as to whether or not the particles 

 move in circles. 



.SECTION D.— Biology. 



Department of Anthropolos;y. 



Address by Francis Galton, F.R.S. 



Permit me to say a few words of personal explanation to 

 account for the form of the .iddress I am about to offer. It has been 

 the custom of my predecessors to give an account of recent pro- 

 ceedings in anthropology, and to touch on many branches of that 

 wide subject. But I am at this moment unprepared to follow 

 their example with the completeness I should desire and you 

 have a right to expect, owing to the suddenness with which I 

 have been called upon to occupy this chair. I had indeed the 

 honour of being nominated to the post last spring, but circum- 

 stances arising which made it highly probable that I should be 

 prevented from attending this meeting, I was compelled to ask 

 to be superseded. New arrangements were then made by the 

 Council, and I thought no more about the matter. However, 

 at the last moment, the accomplished ethnologist who otherwise 

 would have presided over you was liimself debarred by illness 

 from attending, and the original plan had to be reverted to. 



Under these circumstances 1 thought it best to depart some- 

 what from the u&Uil form of addresses, and to confine myself to 

 certain topics with which I happen to have been recently 

 engaged, even at the risk of incurring the charge of submitting 

 to you a memoir rather than an address. 



1 propose to speak of the study of those groups of men who 

 are sufficiently similar in their mental characters or in their 



