JULY 25, 1912] 
NATURE 
529 
German Eigenfrequenz, to designate the natural 
undamped frequency of the system, and, interpreted 
in this sense, the statement to which Prof. Perry 
objects is, with certain limitations, quite correct. 
Apart altogether from the question of nomencla- 
ture, the student finds the usual text-book treatment 
of forced vibrations somewhat unsatisfactory. From 
his dynamics he learns that the amplitude of the 
forced vibration is a maximum when p=W0n?—2k?, 
while the dynamical theory of audition tells him that 
maximum response is excited when p=n. The 
apparent confusion seems to arise from the incom- 
plete differentiation between the two separate varia- 
tions involved—in the limit we may either submit a 
fixed resonator of invariable frequency n/27 to the 
influence of radiation of:yariable wave-length, or we 
may excite a whole range of resonators of different 
frequencies by radiation of invariable frequency p/27. 
In the first case, maximum amplitude is conditioned 
by p=/n?—2k?, and in the second by p=n. These 
results, of course, follow directly from partial differ- 
entiation of the amplitude of the forced vibration 
given by integrating equation (1), but a student is 
more apt to appreciate the distinction between the 
two cases from a geometrical treatment. Write the 
integral of equation (1) in the form 
Bs 5 = A 
Pp he ee ees, | =,5(2) 
where ¢ is the phase angle tan-! Zt 
2k; 
maximum. Draw a line AB to rt ine 4 n*; from it 
cut off BO equal io p*, and at O erect OP perpen- 
dicular to AB and equal to 2kp. When # is constant 
x is obviously a maximum for AO=o, that is, n=. 
If, on the other hand, n is constant, the locus of P 
is a parabola of semiparameter 2k* described with B 
as vertex symmetrically about AB; for a maximum 
value of x in this case the reciprocal of AP must be 
a maximum, hence AP must stand normal to the 
curve, and therefore the subnormal AO must equal 
the semiparameter. Hence n?—p?=2k?. 
Frequently we have to deal with the energy 
absorbed by the resonator, and not at all with the 
amplitude of its forced vibration. The mean rate at 
3 
which energy is absorbed is x sin?¢, so that, for 
2k 
maximum absorption sin?¢=1, or ¢=g90° and n=p 
as before. 
In cases, therefore, where the resonator frequency 
is variable (acoustics, electromagnetic radiation, 
resonance frequency meter, &c.), the condition for 
maximum response is n=, while in cases where the 
frequency of the incident radiation is variable 
(elementary optical theory), the condition for maxi- 
mum absorption is still n=. Where the critical case 
is p=/n?—2k?, we are obviously dealing with a 
maximum opposing force (vibrating spring, ‘ voltage 
resonance,”’ &c.) which is auite different from the case 
of maximum response. 
A very interesting method of examining these 
results is to obtain a series of ‘‘surfaces of ampli- 
tude”’ for various values of the damping factor by 
plotting the two frequencies along perpendicular axes, 
and the amplitude x of the forced vibration along an 
axis perpendicular to the other two. With zero 
damping the surface is symmetrical about the plane 
n=p, and exhibits a ridge rising to infinity in this 
plane. By ascribing any finite value to the damping 
factor k, we occasion a threefold change in the 
character of the surface :— 
NO. 2230, voL. 89] 
1.—The ridge no longer rises to infinity; it 
asymptotically approaches the plane x=o the more 
rapidly the greater the value ascribed to k. 
2.—The symmetrical aspect of the surface is 
destroyed; towards the f, x side of the central plane 
the surface falls lower than towards the other, and at 
the same time the ridge moves towards the n, x side 
of the central plane, the deviation being but slight at 
high frequencies. 
3.—The larger the value ascribed to k, the flatter 
becomes the ridge. The physical interpretation of 
these characteristics is obvious. 
The foregoing results are all well known, but it 
seems advisabie to direct attention to conditions 
which are usually ignored. 
Joun P. Darton. 
University College, Dundee. 
Lobsters in the A€gean. 
Wirn reference to Prof. D’Arcy W. Thompson’s 
letter on this subject in Nature of May 30 (p. 321), it 
may be worth while to record that the British Museum 
(Natural History) has just received fine specimens of 
the common lobster (Homarus gammarus) and the 
spiny lobster or crawfish (Palinurus vulgaris) from 
Smyrna, through the kindness of Capt. J. R. West- 
cott, of the Westcott and Laurance (Ellerman’s) Line, 
Ltd. The existence of both species in the eastern 
Mediterranean is thus confirmed, but it would be of 
great interest, as Prof. Thompson points out, to deter- 
mine the limits of their range and their relative 
abundance in various localities. W. T. Catman. 
British Museum (Natural History), Cromwell Road, 
London, S.W., July 18. 
Wanted—a Flower Sanctuary. 
On revisiting Cheddar last month, after eight 
years, I was horrified to find that plants of Cheddar 
Pink and Thalictrum were being offered for sale to 
visitors. Everyone knows that, when once this sort 
of thing is begun, extermination is only a question 
of time; and in the case of the Cheddar Pink I am 
afraid that it will be a question of a short time only; 
and then this beautiful plant will be lost to the 
English flora. The case is not nearly so serious as 
regards the Thalictrum; but all lovers of Cheddar 
would grieve to see this plant becoming rarer. It is 
important to add, also, that there is just one patch 
of the Welsh Poppy in Cheddar Gorge, and that this 
is so situated that an enterprising dealer could 
exterminate it entirely in a couple of hours or so. I 
am not aware that this plant has yet been taken for 
sale; and it is not a very hopeful subject for trans- 
plantation, perhaps: but, when once the dealer has 
begun his nefarious work, one never knows to what 
lengths he may proceed. I should like, therefore, to 
urge very strongly that any naturalists and nature- 
lovers, who may have any means of bringing influ- 
ence to bear upon the Somersetshire County Council, 
should lose no time in petitioning that body to ‘‘ pro- 
claim” these plants and prohibit their removal, 
especially for sale. There would be no need to inter- 
fere with botanists who gather specimens, or with 
residents and visitors who may wish to take bunches 
of flowers : but what is essential is that the rooting-up 
of the plants should be stopped, and that without loss 
of time. F. H. Prerrycoste. 
Higher Shute Cottage, Polperro, 
Cornwall July 15. 
