TAN 



29 



TAN 



a more musical strain, resembling somewhat, in the mellow- 

 ness of its tones, the sons; of the fifing Baltimore. The 

 syllables to which I have hearkened appear like 'tshooi-e 

 'irnit 'trail, 'rehiiirit trait, and 'irait, 'reftotcit vea wait, 

 with other additions of harmony, for which no words are 

 adequate. This pleasing and highly musical meandering 

 ditty is delivered for hours, in a contemplative mood, in 

 the same tree with his busy consort. If surprised, they 

 flit together, but soon return to their favourite station in 

 the spreading boughs of the shady oak or hickory. This 

 SOUK has some resemblance to that of tin- Red-eyed Vireo 

 in its compass and strain, though much superior, the 'u-att 

 ' being whistled very sweetly in several tones, and with 

 emphasis ; so that, upon the whole, our Pyranga may be 

 considered as duly entitled to various excellencies, being 

 harmless to the farmer, brilliant in plumage, and harmo- 

 nious in voice.' 



Nest, Food, fyc. The same author describes the nest 

 ("which is built about the middle of May, on the horizontal 

 branch of some shady forest-tree, commonly an oak, but 

 sometimes in an orchard tree) as but slightly put together, 

 and usually framed of broken rigid stalks of dry weeds or 

 slender fir-twigs, loosely interlaced together, and partly 

 tied with narrow strips of Indian hemp (Apocy num.), some 

 slender grass-leaves, and pea-vine runners (Amphicarpa), 

 or other frail materials ; the interior being sometimes lined 

 with the slender, wiry, brown stalks of the Canadian cistiis 

 ( Hi'liiiiithi:niii,ii -, or with slender pine-leaves ; the whole 

 so thinly platted as to admit the light through the inter- 

 stices. The three or four eggs are dull blue, spotted with 

 two or three shades of brown or purple, most numerous 

 towards the larger end. As soon as their single brood, 

 which is fledged early in July, is reared, they leave for the 

 south, generally about the middle or end of August. 



' The female, 'says this interesting author in continuation, 

 ' shows great solicitude for the safety of her only brood ; 

 and, on an approach to the nest, appears to be in great dis- 

 tress and apprehension. When they are released from her 

 mure immediate protection, the male, at first cautious and 

 di.stant. now attends and feeds them with activity, being 

 ther indifferent to that concealment which his gaudy 

 oa to require from his natural enemies. So 

 attached to his now interesting brood is the Scarlet Tana- 

 ger, that he has been known, at all hazards, to follow for 

 half a mile one of his young, submitting to feed it atten- 

 tively through the bare of a cage, and. with a devotion 

 which de>pair could not damp, roost by it in the branches 

 of the same tree with its prison." 



The food of this species consists mostly of winged 

 insects, such as wasps, hornets, and wild bees, the smaller 

 kirn! of beetles, and other Coleoplera. Seeds are supposed 

 to be sometimes resorted to, and they are very fond of 

 who'-tle and other benit s. 



It is in August that the moult of the male, when ' he 

 exchanges his nuptial scarlet for the greenish-yellow livery 

 of the female,' commences. (Manual of the Ornithology 

 of the I 'niti-'I States and of Canada.) 



TA.NAGKI'NjE. [TANAGERS.] 

 ' TA'NAIS. [DoN.J 



TANAUO. [Po.] 



TANCRKD, of Hauteville in Normandy, was a feudal 

 baron who lived in the latter part of the tenth and begin- 

 ning of the eleventh century. After doing military service 

 for some years under Richard the Good, duke of Nor- 

 mandy, he retired to his hereditary mansion, where he 

 lived poor, and reared up a numerous family of twelve 

 and three daughters. All his sons were remarkable 

 for their comeliness, their great strength, and their courage. 

 The eldest, Serlon, followed William the Bastard in his 

 conquest of England, and the others went successively to 

 seek their fortune in Apulia, where Rainulf, another Nor- 

 mau adventurer, had already obtained the countship of 

 nn Sergius, duke of Naples. William, one of 

 Tancred's sons, called ' Fier a bras,' or strong of arm, became 

 count of Apulia, and after his death, his brother Robert, 

 called Wiskard, or ' the wise,' became duke of Apulia and 

 >ria, and the founder of the Norman dynasty of Sicily. 

 Two, llntury <>f.~\ Their father Tancred died at 

 a \ cry great age at Hauteville. Traces of the chateau of 

 ied, according to old popular tradition, were still 

 years since in a pretty valley near Hauteville, 

 lour miles Hurt h of the town of Marigny, in the arron 

 uient of Coutances department of La Manche. (Gaultier 



d'Arc, Histoire des Conquftes des Normands en Italie, en 

 Sici/e, et en Grece.) 



TANCRED, son of Eudes, a Norman baron, and of 

 Emma, sister of Robert Wiskard, duke of Apulia, ac- 

 cording to some (Gaultier d'Arc, Histoire des Cofiquctes 

 des Normands en Italie, en Sidle, <c.), and nephew of 

 Bohemund, son of Wiskard, and prince of Tarentum ac- 

 cording to others (Giannone and the authorities he quotes), 

 was serving with Bohemund under Roger, duke of Apulia, 

 son and successor of Wiskard, at the siege of Amalfi, A.D. 

 1096, when the report of the great crusade which was pre- 

 paring for the East determined Bohemund, who was not 

 on good terms with Duke Roger, to join the Crusaders. 

 Tancred followed him with a vast number of men from 

 Apulia and Calabria. The exploits, true or fabulous, of 

 Tancred, in Syria and Palestine, have been immortalized 

 by Tasso in his poem of ' La Gerusalemme.' 



TANCRED, king of Sicily. [SICILIES, Two, History of.) 



TANGENT. In the article CONTACT we have given the 

 first notion on this subject, which we now resume in a 

 somewhat more general manner, annexing the usual de- 

 tails of formulae, but without proof. 



It is usual to apply the word tangent to the tangent 

 straight line only, on which see DIRECTION : generalizing 

 the definition, it will be as follows: Of all curves of a 

 given species, or contained under one equation, that one 

 (B) is the tangent to a given curve (A) at a given point, 

 which passes through that given point, and is nearest to 

 the curve (A) : meaning that no curve of the given species 

 can pass through the given point, so as to pass between 

 (B) and (A), immediately after leaving the point at which 

 the two latter intersect. 



To ascertain the degree of contact of two curves which 

 meet in a point, proceed as follows. Let y = d>v and 

 y=^x be the equations of the curves, and a the abscissa at 

 the point of contact ; so that d>a=^ft. At the point whose 

 abscissa is a+h, the difference of the ordinates of the 

 curves is, by Taylor's theorem, 



h + 



+ (<"'a-,| 



as to which, generally speaking, it will be found that ft 

 can be taken so small that the series shall be convergent : 

 if this be not so, the method of arresting the series given 

 in TAYLOR'S THEOREM must be employed. Now of two 



series of the form AJi"+Bfi" + ____ the value of that in 

 which m is the greaterwill diminish without limit as com- 

 pared with the other, when h diminishes without limit. 

 Consequently, every curve y=$x, which has <f/'</=:^>', will 

 approach, before the point of contact is attained, nearer to 

 y=d>.K than any other in which ty'a is not =d>'a. Again, 

 when d>'a=-j,'a, those cases of y=4/x in which ^"a=d>"a,, 

 will approach nearer to y=0.c than any in which d>"a is 

 not =tj/"a ; and so on. Hence, to make y=i//.r have the 

 closest possible contact with y=(px when x=a ; give such 

 values to the constants in y ^x as will satisfy as many as 

 possible of the equations (f>a=^a,(jifa=^ l a,<f>''a-=^"a, &c. 

 consecutively from the beginning. This is a brief sketch, 

 which can be filled up from any elementary work ; and the 

 following are the principal results : 



1. When the string of equations is satisfied up to 



<jj a=>j/ a, the contact is said to be of the wth order. 



2. In contact of the nth order, the deflection (f>(a+h) 



4/Ca+A; diminishes with h , and vanishes in a finite ratio. 

 to it. 



3. In contact of an even order, the curves intersect at 

 the point of contact ; in contact of an odd order, they do. 

 not intersect at that point. 



4. When curves have a contact of the nth order, no 1 

 curve, having with either a contact of an order inferior to* 

 the nth at the same point, can pass between the two. 



5. A straight line, generally speaking, can have only a. 

 contact of the first order with a curve ; and the equation 

 to the tangent straight line of the curve y=d; when 

 x=a, is y d>a=<p'a(xa). But if it should happen that 



d>"a=0, <'"a = 0, &c., up to <f> W a=0, then for that point 

 the tangent has a contact of the wth order. Thus, at. a 

 point of contrary flexure the tangent has a contact of the 

 second order, at least, with the curve. 



6. A circle, generally speaking, can be made to have a 

 contact of the second order with a curve, and the equation 



