The trill is not obtained by singing to adjacent notes more and more rapidly. Such an attempt will always result in singing two notes, not trilling.
The real trill is a shake. The two tones sound clearly but they are best “TRILLED” by breaking from the preparatory two tones at once into the real shake.
Take exercises 47a and 47b.
Sing the first four notes in moderate speed and try to trill the second measure. There is a slight tension in the larynx when we trill and also a slight oscillation of the larynx, very slight.
In exercise 47b execute the triplets clearly and then attempt the trill again.
If the trill does not come easily be carefully about attempting it many times as it will surely cause hoarseness.
Try it a few times today and again tomorrow, and soon you will probably succeed. It is only fair to say, however, that some people can never learn to trill. If this seems to be your case, give it up until you can be advised by a skillful teacher.
These lessons are designed for home study of the art of singing correctly. Space does not permit including all kinds of musical instruction. Therefore, be sure to study carefully the various turns, ornaments, etc., used in vocal music to designate the will of the composer. All such instruction would require a large book. I feel sure that you are somewhat acquainted with the piano and elementary theory, and I therefore hope that you possess already or will soon obtain a very good knowledge of these subjects. Be sure to become a good musician if you have ambitions to become a real singer. Singers should be just as good musicians as pianists or violinists.
Witherspoon, Herbert. Thirty-Six Lessons in Singing for Teacher and Student. Meissner Institute of Music. Chicago. 1930.